quantum speedup of backtracking and monte carlo …csxam/presentations/...quantum speedup of...
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![Page 1: Quantum speedup of backtracking and Monte Carlo …csxam/presentations/...Quantum speedup of backtracking and Monte Carlo algorithms Ashley Montanaro School of Mathematics, University](https://reader033.vdocuments.us/reader033/viewer/2022041818/5e5c62c5340a5549f72f9c06/html5/thumbnails/1.jpg)
Quantum speedup of backtracking andMonte Carlo algorithms
Ashley Montanaro
School of Mathematics,University of Bristol
19 February 2016
arXiv:1504.06987 and arXiv:1509.02374Proc. R. Soc. A 2015 471 20150301
![Page 2: Quantum speedup of backtracking and Monte Carlo …csxam/presentations/...Quantum speedup of backtracking and Monte Carlo algorithms Ashley Montanaro School of Mathematics, University](https://reader033.vdocuments.us/reader033/viewer/2022041818/5e5c62c5340a5549f72f9c06/html5/thumbnails/2.jpg)
Quantum speedup of classical algorithms
Which general classical algorithmic techniques can we speedup using a quantum computer?
Some examples:
Unstructured search [Grover ’96]
Probability amplification: boost the success probability ofa randomised algorithm to 99% [Brassard et al. ’02]
Probability estimation: determine the success probabilityof a randomised algorithm up to 1% relative error [Brassardet al. ’02]
Simulated annealing [Somma et al. ’07]
In all of these cases, there are quantum algorithms whichachieve quadratic speedups over the corresponding classicalalgorithm.
![Page 3: Quantum speedup of backtracking and Monte Carlo …csxam/presentations/...Quantum speedup of backtracking and Monte Carlo algorithms Ashley Montanaro School of Mathematics, University](https://reader033.vdocuments.us/reader033/viewer/2022041818/5e5c62c5340a5549f72f9c06/html5/thumbnails/3.jpg)
Quantum speedup of classical algorithms
Which general classical algorithmic techniques can we speedup using a quantum computer?
Some examples:
Unstructured search [Grover ’96]
Probability amplification: boost the success probability ofa randomised algorithm to 99% [Brassard et al. ’02]
Probability estimation: determine the success probabilityof a randomised algorithm up to 1% relative error [Brassardet al. ’02]
Simulated annealing [Somma et al. ’07]
In all of these cases, there are quantum algorithms whichachieve quadratic speedups over the corresponding classicalalgorithm.
![Page 4: Quantum speedup of backtracking and Monte Carlo …csxam/presentations/...Quantum speedup of backtracking and Monte Carlo algorithms Ashley Montanaro School of Mathematics, University](https://reader033.vdocuments.us/reader033/viewer/2022041818/5e5c62c5340a5549f72f9c06/html5/thumbnails/4.jpg)
Quantum speedup of classical algorithms
Which general classical algorithmic techniques can we speedup using a quantum computer?
Some examples:
Unstructured search [Grover ’96]
Probability amplification: boost the success probability ofa randomised algorithm to 99% [Brassard et al. ’02]
Probability estimation: determine the success probabilityof a randomised algorithm up to 1% relative error [Brassardet al. ’02]
Simulated annealing [Somma et al. ’07]
In all of these cases, there are quantum algorithms whichachieve quadratic speedups over the corresponding classicalalgorithm.
![Page 5: Quantum speedup of backtracking and Monte Carlo …csxam/presentations/...Quantum speedup of backtracking and Monte Carlo algorithms Ashley Montanaro School of Mathematics, University](https://reader033.vdocuments.us/reader033/viewer/2022041818/5e5c62c5340a5549f72f9c06/html5/thumbnails/5.jpg)
Quantum speedup of classical algorithms
Which general classical algorithmic techniques can we speedup using a quantum computer?
Some examples:
Unstructured search [Grover ’96]
Probability amplification: boost the success probability ofa randomised algorithm to 99% [Brassard et al. ’02]
Probability estimation: determine the success probabilityof a randomised algorithm up to 1% relative error [Brassardet al. ’02]
Simulated annealing [Somma et al. ’07]
In all of these cases, there are quantum algorithms whichachieve quadratic speedups over the corresponding classicalalgorithm.
![Page 6: Quantum speedup of backtracking and Monte Carlo …csxam/presentations/...Quantum speedup of backtracking and Monte Carlo algorithms Ashley Montanaro School of Mathematics, University](https://reader033.vdocuments.us/reader033/viewer/2022041818/5e5c62c5340a5549f72f9c06/html5/thumbnails/6.jpg)
Quantum speedup of classical algorithms
Which general classical algorithmic techniques can we speedup using a quantum computer?
Some examples:
Unstructured search [Grover ’96]
Probability amplification: boost the success probability ofa randomised algorithm to 99% [Brassard et al. ’02]
Probability estimation: determine the success probabilityof a randomised algorithm up to 1% relative error [Brassardet al. ’02]
Simulated annealing [Somma et al. ’07]
In all of these cases, there are quantum algorithms whichachieve quadratic speedups over the corresponding classicalalgorithm.
![Page 7: Quantum speedup of backtracking and Monte Carlo …csxam/presentations/...Quantum speedup of backtracking and Monte Carlo algorithms Ashley Montanaro School of Mathematics, University](https://reader033.vdocuments.us/reader033/viewer/2022041818/5e5c62c5340a5549f72f9c06/html5/thumbnails/7.jpg)
Quantum speedup of classical algorithms
Which general classical algorithmic techniques can we speedup using a quantum computer?
Some examples:
Unstructured search [Grover ’96]
Probability amplification: boost the success probability ofa randomised algorithm to 99% [Brassard et al. ’02]
Probability estimation: determine the success probabilityof a randomised algorithm up to 1% relative error [Brassardet al. ’02]
Simulated annealing [Somma et al. ’07]
In all of these cases, there are quantum algorithms whichachieve quadratic speedups over the corresponding classicalalgorithm.
![Page 8: Quantum speedup of backtracking and Monte Carlo …csxam/presentations/...Quantum speedup of backtracking and Monte Carlo algorithms Ashley Montanaro School of Mathematics, University](https://reader033.vdocuments.us/reader033/viewer/2022041818/5e5c62c5340a5549f72f9c06/html5/thumbnails/8.jpg)
Today’s talk
Two other standard classical algorithmic techniques which canbe accelerated by quantum algorithms:
Backtracking: a standard method for solving constraintsatisfaction problemsApproximating the mean of a random variable withbounded variance: the core of Monte Carlo methods
In both cases, we obtain quadratic quantum speedups.
The quantum algorithms use different techniques:
The backtracking algorithm uses quantum walks, basedon an algorithm of [Belovs ’13].The mean-approximation algorithm uses amplitudeamplification, based on ideas of [Heinrich ’01].
![Page 9: Quantum speedup of backtracking and Monte Carlo …csxam/presentations/...Quantum speedup of backtracking and Monte Carlo algorithms Ashley Montanaro School of Mathematics, University](https://reader033.vdocuments.us/reader033/viewer/2022041818/5e5c62c5340a5549f72f9c06/html5/thumbnails/9.jpg)
Today’s talk
Two other standard classical algorithmic techniques which canbe accelerated by quantum algorithms:
Backtracking: a standard method for solving constraintsatisfaction problems
Approximating the mean of a random variable withbounded variance: the core of Monte Carlo methods
In both cases, we obtain quadratic quantum speedups.
The quantum algorithms use different techniques:
The backtracking algorithm uses quantum walks, basedon an algorithm of [Belovs ’13].The mean-approximation algorithm uses amplitudeamplification, based on ideas of [Heinrich ’01].
![Page 10: Quantum speedup of backtracking and Monte Carlo …csxam/presentations/...Quantum speedup of backtracking and Monte Carlo algorithms Ashley Montanaro School of Mathematics, University](https://reader033.vdocuments.us/reader033/viewer/2022041818/5e5c62c5340a5549f72f9c06/html5/thumbnails/10.jpg)
Today’s talk
Two other standard classical algorithmic techniques which canbe accelerated by quantum algorithms:
Backtracking: a standard method for solving constraintsatisfaction problemsApproximating the mean of a random variable withbounded variance: the core of Monte Carlo methods
In both cases, we obtain quadratic quantum speedups.
The quantum algorithms use different techniques:
The backtracking algorithm uses quantum walks, basedon an algorithm of [Belovs ’13].The mean-approximation algorithm uses amplitudeamplification, based on ideas of [Heinrich ’01].
![Page 11: Quantum speedup of backtracking and Monte Carlo …csxam/presentations/...Quantum speedup of backtracking and Monte Carlo algorithms Ashley Montanaro School of Mathematics, University](https://reader033.vdocuments.us/reader033/viewer/2022041818/5e5c62c5340a5549f72f9c06/html5/thumbnails/11.jpg)
Today’s talk
Two other standard classical algorithmic techniques which canbe accelerated by quantum algorithms:
Backtracking: a standard method for solving constraintsatisfaction problemsApproximating the mean of a random variable withbounded variance: the core of Monte Carlo methods
In both cases, we obtain quadratic quantum speedups.
The quantum algorithms use different techniques:
The backtracking algorithm uses quantum walks, basedon an algorithm of [Belovs ’13].The mean-approximation algorithm uses amplitudeamplification, based on ideas of [Heinrich ’01].
![Page 12: Quantum speedup of backtracking and Monte Carlo …csxam/presentations/...Quantum speedup of backtracking and Monte Carlo algorithms Ashley Montanaro School of Mathematics, University](https://reader033.vdocuments.us/reader033/viewer/2022041818/5e5c62c5340a5549f72f9c06/html5/thumbnails/12.jpg)
Today’s talk
Two other standard classical algorithmic techniques which canbe accelerated by quantum algorithms:
Backtracking: a standard method for solving constraintsatisfaction problemsApproximating the mean of a random variable withbounded variance: the core of Monte Carlo methods
In both cases, we obtain quadratic quantum speedups.
The quantum algorithms use different techniques:
The backtracking algorithm uses quantum walks, basedon an algorithm of [Belovs ’13].The mean-approximation algorithm uses amplitudeamplification, based on ideas of [Heinrich ’01].
![Page 13: Quantum speedup of backtracking and Monte Carlo …csxam/presentations/...Quantum speedup of backtracking and Monte Carlo algorithms Ashley Montanaro School of Mathematics, University](https://reader033.vdocuments.us/reader033/viewer/2022041818/5e5c62c5340a5549f72f9c06/html5/thumbnails/13.jpg)
Constraint satisfaction problems
Backtracking is a general approach to solve constraintsatisfaction problems (CSPs).
An instance of a CSP on n variables x1, . . . , xn is specifiedby a sequence of constraints, all of which must be satisfiedby the variables.
We might want to find one assignment to x1, . . . , xn thatsatisfies all the constraints, or list all such assignments.
For many CSPs, the best algorithms known for either taskhave exponential runtime in n.
A simple example: graph 3-colouring.
![Page 14: Quantum speedup of backtracking and Monte Carlo …csxam/presentations/...Quantum speedup of backtracking and Monte Carlo algorithms Ashley Montanaro School of Mathematics, University](https://reader033.vdocuments.us/reader033/viewer/2022041818/5e5c62c5340a5549f72f9c06/html5/thumbnails/14.jpg)
Constraint satisfaction problems
Backtracking is a general approach to solve constraintsatisfaction problems (CSPs).
An instance of a CSP on n variables x1, . . . , xn is specifiedby a sequence of constraints, all of which must be satisfiedby the variables.
We might want to find one assignment to x1, . . . , xn thatsatisfies all the constraints, or list all such assignments.
For many CSPs, the best algorithms known for either taskhave exponential runtime in n.
A simple example: graph 3-colouring.
![Page 15: Quantum speedup of backtracking and Monte Carlo …csxam/presentations/...Quantum speedup of backtracking and Monte Carlo algorithms Ashley Montanaro School of Mathematics, University](https://reader033.vdocuments.us/reader033/viewer/2022041818/5e5c62c5340a5549f72f9c06/html5/thumbnails/15.jpg)
Constraint satisfaction problems
Backtracking is a general approach to solve constraintsatisfaction problems (CSPs).
An instance of a CSP on n variables x1, . . . , xn is specifiedby a sequence of constraints, all of which must be satisfiedby the variables.
We might want to find one assignment to x1, . . . , xn thatsatisfies all the constraints, or list all such assignments.
For many CSPs, the best algorithms known for either taskhave exponential runtime in n.
A simple example: graph 3-colouring.
![Page 16: Quantum speedup of backtracking and Monte Carlo …csxam/presentations/...Quantum speedup of backtracking and Monte Carlo algorithms Ashley Montanaro School of Mathematics, University](https://reader033.vdocuments.us/reader033/viewer/2022041818/5e5c62c5340a5549f72f9c06/html5/thumbnails/16.jpg)
Constraint satisfaction problems
Backtracking is a general approach to solve constraintsatisfaction problems (CSPs).
An instance of a CSP on n variables x1, . . . , xn is specifiedby a sequence of constraints, all of which must be satisfiedby the variables.
We might want to find one assignment to x1, . . . , xn thatsatisfies all the constraints, or list all such assignments.
For many CSPs, the best algorithms known for either taskhave exponential runtime in n.
A simple example: graph 3-colouring.
![Page 17: Quantum speedup of backtracking and Monte Carlo …csxam/presentations/...Quantum speedup of backtracking and Monte Carlo algorithms Ashley Montanaro School of Mathematics, University](https://reader033.vdocuments.us/reader033/viewer/2022041818/5e5c62c5340a5549f72f9c06/html5/thumbnails/17.jpg)
Constraint satisfaction problems
Backtracking is a general approach to solve constraintsatisfaction problems (CSPs).
An instance of a CSP on n variables x1, . . . , xn is specifiedby a sequence of constraints, all of which must be satisfiedby the variables.
We might want to find one assignment to x1, . . . , xn thatsatisfies all the constraints, or list all such assignments.
For many CSPs, the best algorithms known for either taskhave exponential runtime in n.
A simple example: graph 3-colouring.
![Page 18: Quantum speedup of backtracking and Monte Carlo …csxam/presentations/...Quantum speedup of backtracking and Monte Carlo algorithms Ashley Montanaro School of Mathematics, University](https://reader033.vdocuments.us/reader033/viewer/2022041818/5e5c62c5340a5549f72f9c06/html5/thumbnails/18.jpg)
Constraint satisfaction problems
Backtracking is a general approach to solve constraintsatisfaction problems (CSPs).
An instance of a CSP on n variables x1, . . . , xn is specifiedby a sequence of constraints, all of which must be satisfiedby the variables.
We might want to find one assignment to x1, . . . , xn thatsatisfies all the constraints, or list all such assignments.
For many CSPs, the best algorithms known for either taskhave exponential runtime in n.
A simple example: graph 3-colouring.
![Page 19: Quantum speedup of backtracking and Monte Carlo …csxam/presentations/...Quantum speedup of backtracking and Monte Carlo algorithms Ashley Montanaro School of Mathematics, University](https://reader033.vdocuments.us/reader033/viewer/2022041818/5e5c62c5340a5549f72f9c06/html5/thumbnails/19.jpg)
Colouring by trial and error
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Colouring by trial and error
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Colouring by trial and error
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Colouring by trial and error
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Colouring by trial and error
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Colouring by trial and error
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Colouring by trial and error
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General backtracking framework
This idea, known as backtracking, can be applied to any CSP,given the following assumptions:
We have a problem on n variables, each picked from[d] := 0, . . . , d − 1. Write D := ([d] ∪ ∗)n for the set ofpartial assignments, where ∗ means “not assigned yet”.
We have access to a predicate
P : D→ true, false, indeterminate
which tells us the status of a partial assignment.
We have access to a heuristic
h : D→ 1, . . . ,n
which determines which variable to choose next, for agiven partial assignment.
![Page 27: Quantum speedup of backtracking and Monte Carlo …csxam/presentations/...Quantum speedup of backtracking and Monte Carlo algorithms Ashley Montanaro School of Mathematics, University](https://reader033.vdocuments.us/reader033/viewer/2022041818/5e5c62c5340a5549f72f9c06/html5/thumbnails/27.jpg)
General backtracking framework
This idea, known as backtracking, can be applied to any CSP,given the following assumptions:
We have a problem on n variables, each picked from[d] := 0, . . . , d − 1. Write D := ([d] ∪ ∗)n for the set ofpartial assignments, where ∗ means “not assigned yet”.
We have access to a predicate
P : D→ true, false, indeterminate
which tells us the status of a partial assignment.
We have access to a heuristic
h : D→ 1, . . . ,n
which determines which variable to choose next, for agiven partial assignment.
![Page 28: Quantum speedup of backtracking and Monte Carlo …csxam/presentations/...Quantum speedup of backtracking and Monte Carlo algorithms Ashley Montanaro School of Mathematics, University](https://reader033.vdocuments.us/reader033/viewer/2022041818/5e5c62c5340a5549f72f9c06/html5/thumbnails/28.jpg)
General backtracking framework
This idea, known as backtracking, can be applied to any CSP,given the following assumptions:
We have a problem on n variables, each picked from[d] := 0, . . . , d − 1. Write D := ([d] ∪ ∗)n for the set ofpartial assignments, where ∗ means “not assigned yet”.
We have access to a predicate
P : D→ true, false, indeterminate
which tells us the status of a partial assignment.
We have access to a heuristic
h : D→ 1, . . . ,n
which determines which variable to choose next, for agiven partial assignment.
![Page 29: Quantum speedup of backtracking and Monte Carlo …csxam/presentations/...Quantum speedup of backtracking and Monte Carlo algorithms Ashley Montanaro School of Mathematics, University](https://reader033.vdocuments.us/reader033/viewer/2022041818/5e5c62c5340a5549f72f9c06/html5/thumbnails/29.jpg)
Main result [AM ’15]
TheoremLet T be the number of vertices in the backtracking tree. Thenthere is a bounded-error quantum algorithm which evaluatesP and h O(
√Tn3/2 log n) times each, and outputs x such that
P(x) is true, or “not found” if no such x exists.
If we are promised that there exists a unique x0 such that P(x0)is true, this is improved to O(
√Tn log3 n).
In both cases the algorithm uses poly(n) space and poly(n)auxiliary quantum gates per use of P and h.
The algorithm can be modified to find all solutions bystriking out previously seen solutions.We usually think of T as being exponentially large in n. Inthis regime, this is a near-quadratic separation.Note that the algorithm does not need to know T.
![Page 30: Quantum speedup of backtracking and Monte Carlo …csxam/presentations/...Quantum speedup of backtracking and Monte Carlo algorithms Ashley Montanaro School of Mathematics, University](https://reader033.vdocuments.us/reader033/viewer/2022041818/5e5c62c5340a5549f72f9c06/html5/thumbnails/30.jpg)
Main result [AM ’15]
TheoremLet T be the number of vertices in the backtracking tree. Thenthere is a bounded-error quantum algorithm which evaluatesP and h O(
√Tn3/2 log n) times each, and outputs x such that
P(x) is true, or “not found” if no such x exists.
If we are promised that there exists a unique x0 such that P(x0)is true, this is improved to O(
√Tn log3 n).
In both cases the algorithm uses poly(n) space and poly(n)auxiliary quantum gates per use of P and h.
The algorithm can be modified to find all solutions bystriking out previously seen solutions.We usually think of T as being exponentially large in n. Inthis regime, this is a near-quadratic separation.Note that the algorithm does not need to know T.
![Page 31: Quantum speedup of backtracking and Monte Carlo …csxam/presentations/...Quantum speedup of backtracking and Monte Carlo algorithms Ashley Montanaro School of Mathematics, University](https://reader033.vdocuments.us/reader033/viewer/2022041818/5e5c62c5340a5549f72f9c06/html5/thumbnails/31.jpg)
Main result [AM ’15]
TheoremLet T be the number of vertices in the backtracking tree. Thenthere is a bounded-error quantum algorithm which evaluatesP and h O(
√Tn3/2 log n) times each, and outputs x such that
P(x) is true, or “not found” if no such x exists.
If we are promised that there exists a unique x0 such that P(x0)is true, this is improved to O(
√Tn log3 n).
In both cases the algorithm uses poly(n) space and poly(n)auxiliary quantum gates per use of P and h.
The algorithm can be modified to find all solutions bystriking out previously seen solutions.
We usually think of T as being exponentially large in n. Inthis regime, this is a near-quadratic separation.Note that the algorithm does not need to know T.
![Page 32: Quantum speedup of backtracking and Monte Carlo …csxam/presentations/...Quantum speedup of backtracking and Monte Carlo algorithms Ashley Montanaro School of Mathematics, University](https://reader033.vdocuments.us/reader033/viewer/2022041818/5e5c62c5340a5549f72f9c06/html5/thumbnails/32.jpg)
Main result [AM ’15]
TheoremLet T be the number of vertices in the backtracking tree. Thenthere is a bounded-error quantum algorithm which evaluatesP and h O(
√Tn3/2 log n) times each, and outputs x such that
P(x) is true, or “not found” if no such x exists.
If we are promised that there exists a unique x0 such that P(x0)is true, this is improved to O(
√Tn log3 n).
In both cases the algorithm uses poly(n) space and poly(n)auxiliary quantum gates per use of P and h.
The algorithm can be modified to find all solutions bystriking out previously seen solutions.We usually think of T as being exponentially large in n. Inthis regime, this is a near-quadratic separation.
Note that the algorithm does not need to know T.
![Page 33: Quantum speedup of backtracking and Monte Carlo …csxam/presentations/...Quantum speedup of backtracking and Monte Carlo algorithms Ashley Montanaro School of Mathematics, University](https://reader033.vdocuments.us/reader033/viewer/2022041818/5e5c62c5340a5549f72f9c06/html5/thumbnails/33.jpg)
Main result [AM ’15]
TheoremLet T be the number of vertices in the backtracking tree. Thenthere is a bounded-error quantum algorithm which evaluatesP and h O(
√Tn3/2 log n) times each, and outputs x such that
P(x) is true, or “not found” if no such x exists.
If we are promised that there exists a unique x0 such that P(x0)is true, this is improved to O(
√Tn log3 n).
In both cases the algorithm uses poly(n) space and poly(n)auxiliary quantum gates per use of P and h.
The algorithm can be modified to find all solutions bystriking out previously seen solutions.We usually think of T as being exponentially large in n. Inthis regime, this is a near-quadratic separation.Note that the algorithm does not need to know T.
![Page 34: Quantum speedup of backtracking and Monte Carlo …csxam/presentations/...Quantum speedup of backtracking and Monte Carlo algorithms Ashley Montanaro School of Mathematics, University](https://reader033.vdocuments.us/reader033/viewer/2022041818/5e5c62c5340a5549f72f9c06/html5/thumbnails/34.jpg)
Previous work
Some previous works have developed quantum algorithmsrelated to backtracking:
[Cerf, Grover and Williams ’00] developed a quantumalgorithm for constraint satisfaction problems, based on anested version of Grover search. This can be seen as aquantum version of one particular backtracking algorithmthat runs quadratically faster.
[Farhi and Gutmann ’98] used continuous-time quantumwalks to find solutions in backtracking trees. Theyshowed that, for some trees, the quantum walk can find asolution exponentially faster than a classical random walk.
By contrast, the algorithm presented here achieves a (nearly)quadratic separation for all trees.
![Page 35: Quantum speedup of backtracking and Monte Carlo …csxam/presentations/...Quantum speedup of backtracking and Monte Carlo algorithms Ashley Montanaro School of Mathematics, University](https://reader033.vdocuments.us/reader033/viewer/2022041818/5e5c62c5340a5549f72f9c06/html5/thumbnails/35.jpg)
Previous work
Some previous works have developed quantum algorithmsrelated to backtracking:
[Cerf, Grover and Williams ’00] developed a quantumalgorithm for constraint satisfaction problems, based on anested version of Grover search. This can be seen as aquantum version of one particular backtracking algorithmthat runs quadratically faster.
[Farhi and Gutmann ’98] used continuous-time quantumwalks to find solutions in backtracking trees. Theyshowed that, for some trees, the quantum walk can find asolution exponentially faster than a classical random walk.
By contrast, the algorithm presented here achieves a (nearly)quadratic separation for all trees.
![Page 36: Quantum speedup of backtracking and Monte Carlo …csxam/presentations/...Quantum speedup of backtracking and Monte Carlo algorithms Ashley Montanaro School of Mathematics, University](https://reader033.vdocuments.us/reader033/viewer/2022041818/5e5c62c5340a5549f72f9c06/html5/thumbnails/36.jpg)
Previous work
Some previous works have developed quantum algorithmsrelated to backtracking:
[Cerf, Grover and Williams ’00] developed a quantumalgorithm for constraint satisfaction problems, based on anested version of Grover search. This can be seen as aquantum version of one particular backtracking algorithmthat runs quadratically faster.
[Farhi and Gutmann ’98] used continuous-time quantumwalks to find solutions in backtracking trees. Theyshowed that, for some trees, the quantum walk can find asolution exponentially faster than a classical random walk.
By contrast, the algorithm presented here achieves a (nearly)quadratic separation for all trees.
![Page 37: Quantum speedup of backtracking and Monte Carlo …csxam/presentations/...Quantum speedup of backtracking and Monte Carlo algorithms Ashley Montanaro School of Mathematics, University](https://reader033.vdocuments.us/reader033/viewer/2022041818/5e5c62c5340a5549f72f9c06/html5/thumbnails/37.jpg)
Search in the backtracking tree
Idea: Use quantum search to find a solution (“marked vertex”)in the tree produced by the backtracking algorithm.
Many works have studied quantum search in various graphs,e.g. [Szegedy ’04], [Aaronson and Ambainis ’05], [Magniez et al. ’11] . . .
But here there are some difficulties:The graph is not known in advance, and is determined bythe backtracking algorithm.We start at the root of the tree, not in the stationarydistribution of a random walk on the graph.
These can be overcome using work of [Belovs ’13] relatingquantum walks to effective resistance in an electrical network.
![Page 38: Quantum speedup of backtracking and Monte Carlo …csxam/presentations/...Quantum speedup of backtracking and Monte Carlo algorithms Ashley Montanaro School of Mathematics, University](https://reader033.vdocuments.us/reader033/viewer/2022041818/5e5c62c5340a5549f72f9c06/html5/thumbnails/38.jpg)
Search in the backtracking tree
Idea: Use quantum search to find a solution (“marked vertex”)in the tree produced by the backtracking algorithm.
Many works have studied quantum search in various graphs,e.g. [Szegedy ’04], [Aaronson and Ambainis ’05], [Magniez et al. ’11] . . .
But here there are some difficulties:The graph is not known in advance, and is determined bythe backtracking algorithm.We start at the root of the tree, not in the stationarydistribution of a random walk on the graph.
These can be overcome using work of [Belovs ’13] relatingquantum walks to effective resistance in an electrical network.
![Page 39: Quantum speedup of backtracking and Monte Carlo …csxam/presentations/...Quantum speedup of backtracking and Monte Carlo algorithms Ashley Montanaro School of Mathematics, University](https://reader033.vdocuments.us/reader033/viewer/2022041818/5e5c62c5340a5549f72f9c06/html5/thumbnails/39.jpg)
Search in the backtracking tree
Idea: Use quantum search to find a solution (“marked vertex”)in the tree produced by the backtracking algorithm.
Many works have studied quantum search in various graphs,e.g. [Szegedy ’04], [Aaronson and Ambainis ’05], [Magniez et al. ’11] . . .
But here there are some difficulties:The graph is not known in advance, and is determined bythe backtracking algorithm.
We start at the root of the tree, not in the stationarydistribution of a random walk on the graph.
These can be overcome using work of [Belovs ’13] relatingquantum walks to effective resistance in an electrical network.
![Page 40: Quantum speedup of backtracking and Monte Carlo …csxam/presentations/...Quantum speedup of backtracking and Monte Carlo algorithms Ashley Montanaro School of Mathematics, University](https://reader033.vdocuments.us/reader033/viewer/2022041818/5e5c62c5340a5549f72f9c06/html5/thumbnails/40.jpg)
Search in the backtracking tree
Idea: Use quantum search to find a solution (“marked vertex”)in the tree produced by the backtracking algorithm.
Many works have studied quantum search in various graphs,e.g. [Szegedy ’04], [Aaronson and Ambainis ’05], [Magniez et al. ’11] . . .
But here there are some difficulties:The graph is not known in advance, and is determined bythe backtracking algorithm.We start at the root of the tree, not in the stationarydistribution of a random walk on the graph.
These can be overcome using work of [Belovs ’13] relatingquantum walks to effective resistance in an electrical network.
![Page 41: Quantum speedup of backtracking and Monte Carlo …csxam/presentations/...Quantum speedup of backtracking and Monte Carlo algorithms Ashley Montanaro School of Mathematics, University](https://reader033.vdocuments.us/reader033/viewer/2022041818/5e5c62c5340a5549f72f9c06/html5/thumbnails/41.jpg)
Search in the backtracking tree
Idea: Use quantum search to find a solution (“marked vertex”)in the tree produced by the backtracking algorithm.
Many works have studied quantum search in various graphs,e.g. [Szegedy ’04], [Aaronson and Ambainis ’05], [Magniez et al. ’11] . . .
But here there are some difficulties:The graph is not known in advance, and is determined bythe backtracking algorithm.We start at the root of the tree, not in the stationarydistribution of a random walk on the graph.
These can be overcome using work of [Belovs ’13] relatingquantum walks to effective resistance in an electrical network.
![Page 42: Quantum speedup of backtracking and Monte Carlo …csxam/presentations/...Quantum speedup of backtracking and Monte Carlo algorithms Ashley Montanaro School of Mathematics, University](https://reader033.vdocuments.us/reader033/viewer/2022041818/5e5c62c5340a5549f72f9c06/html5/thumbnails/42.jpg)
Search by quantum walk (sketch)
We apply phase estimation to a quantum walk starting at theroot, with precision O(1/
√Tn), where n is an upper bound on
the depth of the tree, and output “solution exists” if theeigenvalue is 1, and “no solution” otherwise.
Claim (special case of [Belovs ’13])This procedure succeeds with probability O(1).
So we can detect the existence of a solution with O(√
Tn)quantum walk steps.Each quantum walk step can be implemented with O(1)uses of P and h.We can also find a solution using binary search with asmall overhead.
![Page 43: Quantum speedup of backtracking and Monte Carlo …csxam/presentations/...Quantum speedup of backtracking and Monte Carlo algorithms Ashley Montanaro School of Mathematics, University](https://reader033.vdocuments.us/reader033/viewer/2022041818/5e5c62c5340a5549f72f9c06/html5/thumbnails/43.jpg)
Search by quantum walk (sketch)
We apply phase estimation to a quantum walk starting at theroot, with precision O(1/
√Tn), where n is an upper bound on
the depth of the tree, and output “solution exists” if theeigenvalue is 1, and “no solution” otherwise.
Claim (special case of [Belovs ’13])This procedure succeeds with probability O(1).
So we can detect the existence of a solution with O(√
Tn)quantum walk steps.Each quantum walk step can be implemented with O(1)uses of P and h.We can also find a solution using binary search with asmall overhead.
![Page 44: Quantum speedup of backtracking and Monte Carlo …csxam/presentations/...Quantum speedup of backtracking and Monte Carlo algorithms Ashley Montanaro School of Mathematics, University](https://reader033.vdocuments.us/reader033/viewer/2022041818/5e5c62c5340a5549f72f9c06/html5/thumbnails/44.jpg)
Search by quantum walk (sketch)
We apply phase estimation to a quantum walk starting at theroot, with precision O(1/
√Tn), where n is an upper bound on
the depth of the tree, and output “solution exists” if theeigenvalue is 1, and “no solution” otherwise.
Claim (special case of [Belovs ’13])This procedure succeeds with probability O(1).
So we can detect the existence of a solution with O(√
Tn)quantum walk steps.
Each quantum walk step can be implemented with O(1)uses of P and h.We can also find a solution using binary search with asmall overhead.
![Page 45: Quantum speedup of backtracking and Monte Carlo …csxam/presentations/...Quantum speedup of backtracking and Monte Carlo algorithms Ashley Montanaro School of Mathematics, University](https://reader033.vdocuments.us/reader033/viewer/2022041818/5e5c62c5340a5549f72f9c06/html5/thumbnails/45.jpg)
Search by quantum walk (sketch)
We apply phase estimation to a quantum walk starting at theroot, with precision O(1/
√Tn), where n is an upper bound on
the depth of the tree, and output “solution exists” if theeigenvalue is 1, and “no solution” otherwise.
Claim (special case of [Belovs ’13])This procedure succeeds with probability O(1).
So we can detect the existence of a solution with O(√
Tn)quantum walk steps.Each quantum walk step can be implemented with O(1)uses of P and h.
We can also find a solution using binary search with asmall overhead.
![Page 46: Quantum speedup of backtracking and Monte Carlo …csxam/presentations/...Quantum speedup of backtracking and Monte Carlo algorithms Ashley Montanaro School of Mathematics, University](https://reader033.vdocuments.us/reader033/viewer/2022041818/5e5c62c5340a5549f72f9c06/html5/thumbnails/46.jpg)
Search by quantum walk (sketch)
We apply phase estimation to a quantum walk starting at theroot, with precision O(1/
√Tn), where n is an upper bound on
the depth of the tree, and output “solution exists” if theeigenvalue is 1, and “no solution” otherwise.
Claim (special case of [Belovs ’13])This procedure succeeds with probability O(1).
So we can detect the existence of a solution with O(√
Tn)quantum walk steps.Each quantum walk step can be implemented with O(1)uses of P and h.We can also find a solution using binary search with asmall overhead.
![Page 47: Quantum speedup of backtracking and Monte Carlo …csxam/presentations/...Quantum speedup of backtracking and Monte Carlo algorithms Ashley Montanaro School of Mathematics, University](https://reader033.vdocuments.us/reader033/viewer/2022041818/5e5c62c5340a5549f72f9c06/html5/thumbnails/47.jpg)
Part 2: Monte Carlo methodsMonte Carlo methods use randomness to estimate numericalproperties of systems which are too large or complicated toanalyse deterministically.
Pic: Wikipedia
![Page 48: Quantum speedup of backtracking and Monte Carlo …csxam/presentations/...Quantum speedup of backtracking and Monte Carlo algorithms Ashley Montanaro School of Mathematics, University](https://reader033.vdocuments.us/reader033/viewer/2022041818/5e5c62c5340a5549f72f9c06/html5/thumbnails/48.jpg)
Part 2: Monte Carlo methodsMonte Carlo methods use randomness to estimate numericalproperties of systems which are too large or complicated toanalyse deterministically.
Pic: Wikipedia
![Page 49: Quantum speedup of backtracking and Monte Carlo …csxam/presentations/...Quantum speedup of backtracking and Monte Carlo algorithms Ashley Montanaro School of Mathematics, University](https://reader033.vdocuments.us/reader033/viewer/2022041818/5e5c62c5340a5549f72f9c06/html5/thumbnails/49.jpg)
These methods are used throughout science and engineering:
![Page 50: Quantum speedup of backtracking and Monte Carlo …csxam/presentations/...Quantum speedup of backtracking and Monte Carlo algorithms Ashley Montanaro School of Mathematics, University](https://reader033.vdocuments.us/reader033/viewer/2022041818/5e5c62c5340a5549f72f9c06/html5/thumbnails/50.jpg)
. . . and were an application of the first electronic computers:
Pic: Wikipedia
![Page 51: Quantum speedup of backtracking and Monte Carlo …csxam/presentations/...Quantum speedup of backtracking and Monte Carlo algorithms Ashley Montanaro School of Mathematics, University](https://reader033.vdocuments.us/reader033/viewer/2022041818/5e5c62c5340a5549f72f9c06/html5/thumbnails/51.jpg)
Monte Carlo methods
The basic core of many Monte Carlo methods is:
General problemGiven access to a randomised algorithm A, estimate theexpected output value µ of A.
The input is fixed, and the expectation is taken over theinternal randomness of A.
The output value v(A) is a real-valued random variable.
We assume that we know an upper bound on the variance ofthis random variable:
Var(v(A)) 6 σ2.
![Page 52: Quantum speedup of backtracking and Monte Carlo …csxam/presentations/...Quantum speedup of backtracking and Monte Carlo algorithms Ashley Montanaro School of Mathematics, University](https://reader033.vdocuments.us/reader033/viewer/2022041818/5e5c62c5340a5549f72f9c06/html5/thumbnails/52.jpg)
Monte Carlo methods
The basic core of many Monte Carlo methods is:
General problemGiven access to a randomised algorithm A, estimate theexpected output value µ of A.
The input is fixed, and the expectation is taken over theinternal randomness of A.
The output value v(A) is a real-valued random variable.
We assume that we know an upper bound on the variance ofthis random variable:
Var(v(A)) 6 σ2.
![Page 53: Quantum speedup of backtracking and Monte Carlo …csxam/presentations/...Quantum speedup of backtracking and Monte Carlo algorithms Ashley Montanaro School of Mathematics, University](https://reader033.vdocuments.us/reader033/viewer/2022041818/5e5c62c5340a5549f72f9c06/html5/thumbnails/53.jpg)
Monte Carlo methods
The basic core of many Monte Carlo methods is:
General problemGiven access to a randomised algorithm A, estimate theexpected output value µ of A.
The input is fixed, and the expectation is taken over theinternal randomness of A.
The output value v(A) is a real-valued random variable.
We assume that we know an upper bound on the variance ofthis random variable:
Var(v(A)) 6 σ2.
![Page 54: Quantum speedup of backtracking and Monte Carlo …csxam/presentations/...Quantum speedup of backtracking and Monte Carlo algorithms Ashley Montanaro School of Mathematics, University](https://reader033.vdocuments.us/reader033/viewer/2022041818/5e5c62c5340a5549f72f9c06/html5/thumbnails/54.jpg)
Classical algorithm
The following natural algorithm solves this problem for any A:
1 Produce k samples v1, . . . , vk, each corresponding to theoutput of an independent execution of A.
2 Output the average µ = 1k∑k
i=1 vi of the samples as anapproximation of µ.
Assuming that the variance of v(A) is at most σ2,
Pr[|µ− µ| > ε] 6σ2
kε2 .
So we can take k = O(σ2/ε2) to estimate µ up to additive errorε with, say, 99% success probability.
This scaling is optimal for classical algorithms [Dagum et al. ’00].
![Page 55: Quantum speedup of backtracking and Monte Carlo …csxam/presentations/...Quantum speedup of backtracking and Monte Carlo algorithms Ashley Montanaro School of Mathematics, University](https://reader033.vdocuments.us/reader033/viewer/2022041818/5e5c62c5340a5549f72f9c06/html5/thumbnails/55.jpg)
Classical algorithm
The following natural algorithm solves this problem for any A:
1 Produce k samples v1, . . . , vk, each corresponding to theoutput of an independent execution of A.
2 Output the average µ = 1k∑k
i=1 vi of the samples as anapproximation of µ.
Assuming that the variance of v(A) is at most σ2,
Pr[|µ− µ| > ε] 6σ2
kε2 .
So we can take k = O(σ2/ε2) to estimate µ up to additive errorε with, say, 99% success probability.
This scaling is optimal for classical algorithms [Dagum et al. ’00].
![Page 56: Quantum speedup of backtracking and Monte Carlo …csxam/presentations/...Quantum speedup of backtracking and Monte Carlo algorithms Ashley Montanaro School of Mathematics, University](https://reader033.vdocuments.us/reader033/viewer/2022041818/5e5c62c5340a5549f72f9c06/html5/thumbnails/56.jpg)
Classical algorithm
The following natural algorithm solves this problem for any A:
1 Produce k samples v1, . . . , vk, each corresponding to theoutput of an independent execution of A.
2 Output the average µ = 1k∑k
i=1 vi of the samples as anapproximation of µ.
Assuming that the variance of v(A) is at most σ2,
Pr[|µ− µ| > ε] 6σ2
kε2 .
So we can take k = O(σ2/ε2) to estimate µ up to additive errorε with, say, 99% success probability.
This scaling is optimal for classical algorithms [Dagum et al. ’00].
![Page 57: Quantum speedup of backtracking and Monte Carlo …csxam/presentations/...Quantum speedup of backtracking and Monte Carlo algorithms Ashley Montanaro School of Mathematics, University](https://reader033.vdocuments.us/reader033/viewer/2022041818/5e5c62c5340a5549f72f9c06/html5/thumbnails/57.jpg)
Classical algorithm
The following natural algorithm solves this problem for any A:
1 Produce k samples v1, . . . , vk, each corresponding to theoutput of an independent execution of A.
2 Output the average µ = 1k∑k
i=1 vi of the samples as anapproximation of µ.
Assuming that the variance of v(A) is at most σ2,
Pr[|µ− µ| > ε] 6σ2
kε2 .
So we can take k = O(σ2/ε2) to estimate µ up to additive errorε with, say, 99% success probability.
This scaling is optimal for classical algorithms [Dagum et al. ’00].
![Page 58: Quantum speedup of backtracking and Monte Carlo …csxam/presentations/...Quantum speedup of backtracking and Monte Carlo algorithms Ashley Montanaro School of Mathematics, University](https://reader033.vdocuments.us/reader033/viewer/2022041818/5e5c62c5340a5549f72f9c06/html5/thumbnails/58.jpg)
Quantum speedup
With a quantum computer, we can do better:
Theorem [AM ’15]
There is a quantum algorithm which estimates µ up toadditive error ε with 99% success probability and
O(σ/ε)
uses of A (and A−1).
The O notation hides polylog factors: more precisely, thecomplexity is O((σ/ε) log3/2(σ/ε) log log(σ/ε)).
This complexity is optimal up to these polylog factors[Nayak and Wu ’98].
The underlying algorithm A can now be quantum itself.
![Page 59: Quantum speedup of backtracking and Monte Carlo …csxam/presentations/...Quantum speedup of backtracking and Monte Carlo algorithms Ashley Montanaro School of Mathematics, University](https://reader033.vdocuments.us/reader033/viewer/2022041818/5e5c62c5340a5549f72f9c06/html5/thumbnails/59.jpg)
Quantum speedup
With a quantum computer, we can do better:
Theorem [AM ’15]
There is a quantum algorithm which estimates µ up toadditive error ε with 99% success probability and
O(σ/ε)
uses of A (and A−1).
The O notation hides polylog factors: more precisely, thecomplexity is O((σ/ε) log3/2(σ/ε) log log(σ/ε)).
This complexity is optimal up to these polylog factors[Nayak and Wu ’98].
The underlying algorithm A can now be quantum itself.
![Page 60: Quantum speedup of backtracking and Monte Carlo …csxam/presentations/...Quantum speedup of backtracking and Monte Carlo algorithms Ashley Montanaro School of Mathematics, University](https://reader033.vdocuments.us/reader033/viewer/2022041818/5e5c62c5340a5549f72f9c06/html5/thumbnails/60.jpg)
Quantum speedup
With a quantum computer, we can do better:
Theorem [AM ’15]
There is a quantum algorithm which estimates µ up toadditive error ε with 99% success probability and
O(σ/ε)
uses of A (and A−1).
The O notation hides polylog factors: more precisely, thecomplexity is O((σ/ε) log3/2(σ/ε) log log(σ/ε)).
This complexity is optimal up to these polylog factors[Nayak and Wu ’98].
The underlying algorithm A can now be quantum itself.
![Page 61: Quantum speedup of backtracking and Monte Carlo …csxam/presentations/...Quantum speedup of backtracking and Monte Carlo algorithms Ashley Montanaro School of Mathematics, University](https://reader033.vdocuments.us/reader033/viewer/2022041818/5e5c62c5340a5549f72f9c06/html5/thumbnails/61.jpg)
Quantum speedup
With a quantum computer, we can do better:
Theorem [AM ’15]
There is a quantum algorithm which estimates µ up toadditive error ε with 99% success probability and
O(σ/ε)
uses of A (and A−1).
The O notation hides polylog factors: more precisely, thecomplexity is O((σ/ε) log3/2(σ/ε) log log(σ/ε)).
This complexity is optimal up to these polylog factors[Nayak and Wu ’98].
The underlying algorithm A can now be quantum itself.
![Page 62: Quantum speedup of backtracking and Monte Carlo …csxam/presentations/...Quantum speedup of backtracking and Monte Carlo algorithms Ashley Montanaro School of Mathematics, University](https://reader033.vdocuments.us/reader033/viewer/2022041818/5e5c62c5340a5549f72f9c06/html5/thumbnails/62.jpg)
Related work
This problem connects to several previous works, e.g.:
Approximating the mean of an arbitrary boundedfunction (with range [0, 1]), with respect to the uniformdistribution. Quantum complexity: O(1/ε) [Heinrich ’01],[Brassard et al. ’11].
Estimating the expected value tr(Aρ) of certainobservables A which are bounded [Wocjan et al. ’09], orwhose tails decay quickly [Knill, Ortiz and Somma ’07].
Approximating the mean, with respect to the uniformdistribution, of functions with bounded L2 norm [Heinrich’01]
Here we generalise these by approximating the mean outputvalue of arbitrary quantum algorithms, given only a bound onthe variance.
![Page 63: Quantum speedup of backtracking and Monte Carlo …csxam/presentations/...Quantum speedup of backtracking and Monte Carlo algorithms Ashley Montanaro School of Mathematics, University](https://reader033.vdocuments.us/reader033/viewer/2022041818/5e5c62c5340a5549f72f9c06/html5/thumbnails/63.jpg)
Related work
This problem connects to several previous works, e.g.:
Approximating the mean of an arbitrary boundedfunction (with range [0, 1]), with respect to the uniformdistribution. Quantum complexity: O(1/ε) [Heinrich ’01],[Brassard et al. ’11].
Estimating the expected value tr(Aρ) of certainobservables A which are bounded [Wocjan et al. ’09], orwhose tails decay quickly [Knill, Ortiz and Somma ’07].
Approximating the mean, with respect to the uniformdistribution, of functions with bounded L2 norm [Heinrich’01]
Here we generalise these by approximating the mean outputvalue of arbitrary quantum algorithms, given only a bound onthe variance.
![Page 64: Quantum speedup of backtracking and Monte Carlo …csxam/presentations/...Quantum speedup of backtracking and Monte Carlo algorithms Ashley Montanaro School of Mathematics, University](https://reader033.vdocuments.us/reader033/viewer/2022041818/5e5c62c5340a5549f72f9c06/html5/thumbnails/64.jpg)
Related work
This problem connects to several previous works, e.g.:
Approximating the mean of an arbitrary boundedfunction (with range [0, 1]), with respect to the uniformdistribution. Quantum complexity: O(1/ε) [Heinrich ’01],[Brassard et al. ’11].
Estimating the expected value tr(Aρ) of certainobservables A which are bounded [Wocjan et al. ’09], orwhose tails decay quickly [Knill, Ortiz and Somma ’07].
Approximating the mean, with respect to the uniformdistribution, of functions with bounded L2 norm [Heinrich’01]
Here we generalise these by approximating the mean outputvalue of arbitrary quantum algorithms, given only a bound onthe variance.
![Page 65: Quantum speedup of backtracking and Monte Carlo …csxam/presentations/...Quantum speedup of backtracking and Monte Carlo algorithms Ashley Montanaro School of Mathematics, University](https://reader033.vdocuments.us/reader033/viewer/2022041818/5e5c62c5340a5549f72f9c06/html5/thumbnails/65.jpg)
Related work
This problem connects to several previous works, e.g.:
Approximating the mean of an arbitrary boundedfunction (with range [0, 1]), with respect to the uniformdistribution. Quantum complexity: O(1/ε) [Heinrich ’01],[Brassard et al. ’11].
Estimating the expected value tr(Aρ) of certainobservables A which are bounded [Wocjan et al. ’09], orwhose tails decay quickly [Knill, Ortiz and Somma ’07].
Approximating the mean, with respect to the uniformdistribution, of functions with bounded L2 norm [Heinrich’01]
Here we generalise these by approximating the mean outputvalue of arbitrary quantum algorithms, given only a bound onthe variance.
![Page 66: Quantum speedup of backtracking and Monte Carlo …csxam/presentations/...Quantum speedup of backtracking and Monte Carlo algorithms Ashley Montanaro School of Mathematics, University](https://reader033.vdocuments.us/reader033/viewer/2022041818/5e5c62c5340a5549f72f9c06/html5/thumbnails/66.jpg)
Ideas behind the algorithm
The algorithm combines and extends ideas of [Heinrich ’01],[Brassard et al. ’11], [Wocjan et al. ’09].
A sketch of the argument:
If we know that the output of A is bounded in [0, 1], wecan use amplitude estimation to approximate µ up to ε,using A (and A−1) O(1/ε) times.
So divide up the output values of A into blocks ofexponentially increasing distance from µ.
Rescale and shift the values in each block to be boundedin [0, 1]. Then use amplitude estimation to estimate theaverage output value in each block.
Sum up the results (after rescaling them again).
This works because, if the variance of A is low, output valuesfar from µ do not contribute much to µ, so can be estimatedwith lower precision.
![Page 67: Quantum speedup of backtracking and Monte Carlo …csxam/presentations/...Quantum speedup of backtracking and Monte Carlo algorithms Ashley Montanaro School of Mathematics, University](https://reader033.vdocuments.us/reader033/viewer/2022041818/5e5c62c5340a5549f72f9c06/html5/thumbnails/67.jpg)
Ideas behind the algorithm
The algorithm combines and extends ideas of [Heinrich ’01],[Brassard et al. ’11], [Wocjan et al. ’09]. A sketch of the argument:
If we know that the output of A is bounded in [0, 1], wecan use amplitude estimation to approximate µ up to ε,using A (and A−1) O(1/ε) times.
So divide up the output values of A into blocks ofexponentially increasing distance from µ.
Rescale and shift the values in each block to be boundedin [0, 1]. Then use amplitude estimation to estimate theaverage output value in each block.
Sum up the results (after rescaling them again).
This works because, if the variance of A is low, output valuesfar from µ do not contribute much to µ, so can be estimatedwith lower precision.
![Page 68: Quantum speedup of backtracking and Monte Carlo …csxam/presentations/...Quantum speedup of backtracking and Monte Carlo algorithms Ashley Montanaro School of Mathematics, University](https://reader033.vdocuments.us/reader033/viewer/2022041818/5e5c62c5340a5549f72f9c06/html5/thumbnails/68.jpg)
Ideas behind the algorithm
The algorithm combines and extends ideas of [Heinrich ’01],[Brassard et al. ’11], [Wocjan et al. ’09]. A sketch of the argument:
If we know that the output of A is bounded in [0, 1], wecan use amplitude estimation to approximate µ up to ε,using A (and A−1) O(1/ε) times.
So divide up the output values of A into blocks ofexponentially increasing distance from µ.
Rescale and shift the values in each block to be boundedin [0, 1]. Then use amplitude estimation to estimate theaverage output value in each block.
Sum up the results (after rescaling them again).
This works because, if the variance of A is low, output valuesfar from µ do not contribute much to µ, so can be estimatedwith lower precision.
![Page 69: Quantum speedup of backtracking and Monte Carlo …csxam/presentations/...Quantum speedup of backtracking and Monte Carlo algorithms Ashley Montanaro School of Mathematics, University](https://reader033.vdocuments.us/reader033/viewer/2022041818/5e5c62c5340a5549f72f9c06/html5/thumbnails/69.jpg)
Ideas behind the algorithm
The algorithm combines and extends ideas of [Heinrich ’01],[Brassard et al. ’11], [Wocjan et al. ’09]. A sketch of the argument:
If we know that the output of A is bounded in [0, 1], wecan use amplitude estimation to approximate µ up to ε,using A (and A−1) O(1/ε) times.
So divide up the output values of A into blocks ofexponentially increasing distance from µ.
Rescale and shift the values in each block to be boundedin [0, 1]. Then use amplitude estimation to estimate theaverage output value in each block.
Sum up the results (after rescaling them again).
This works because, if the variance of A is low, output valuesfar from µ do not contribute much to µ, so can be estimatedwith lower precision.
![Page 70: Quantum speedup of backtracking and Monte Carlo …csxam/presentations/...Quantum speedup of backtracking and Monte Carlo algorithms Ashley Montanaro School of Mathematics, University](https://reader033.vdocuments.us/reader033/viewer/2022041818/5e5c62c5340a5549f72f9c06/html5/thumbnails/70.jpg)
Ideas behind the algorithm
The algorithm combines and extends ideas of [Heinrich ’01],[Brassard et al. ’11], [Wocjan et al. ’09]. A sketch of the argument:
If we know that the output of A is bounded in [0, 1], wecan use amplitude estimation to approximate µ up to ε,using A (and A−1) O(1/ε) times.
So divide up the output values of A into blocks ofexponentially increasing distance from µ.
Rescale and shift the values in each block to be boundedin [0, 1]. Then use amplitude estimation to estimate theaverage output value in each block.
Sum up the results (after rescaling them again).
This works because, if the variance of A is low, output valuesfar from µ do not contribute much to µ, so can be estimatedwith lower precision.
![Page 71: Quantum speedup of backtracking and Monte Carlo …csxam/presentations/...Quantum speedup of backtracking and Monte Carlo algorithms Ashley Montanaro School of Mathematics, University](https://reader033.vdocuments.us/reader033/viewer/2022041818/5e5c62c5340a5549f72f9c06/html5/thumbnails/71.jpg)
Ideas behind the algorithm
The algorithm combines and extends ideas of [Heinrich ’01],[Brassard et al. ’11], [Wocjan et al. ’09]. A sketch of the argument:
If we know that the output of A is bounded in [0, 1], wecan use amplitude estimation to approximate µ up to ε,using A (and A−1) O(1/ε) times.
So divide up the output values of A into blocks ofexponentially increasing distance from µ.
Rescale and shift the values in each block to be boundedin [0, 1]. Then use amplitude estimation to estimate theaverage output value in each block.
Sum up the results (after rescaling them again).
This works because, if the variance of A is low, output valuesfar from µ do not contribute much to µ, so can be estimatedwith lower precision.
![Page 72: Quantum speedup of backtracking and Monte Carlo …csxam/presentations/...Quantum speedup of backtracking and Monte Carlo algorithms Ashley Montanaro School of Mathematics, University](https://reader033.vdocuments.us/reader033/viewer/2022041818/5e5c62c5340a5549f72f9c06/html5/thumbnails/72.jpg)
Application: partition functions
Consider a (classical) physical system which has state space Ω,and a Hamiltonian H : Ω→ R specifying the energy of eachconfiguration x ∈ Ω. Assume that H takes integer values in theset 0, . . . ,n.
We want to compute the partition function
Z(β) =∑x∈Ω
e−βH(x)
for some inverse temperature β.
Encapsulates some interesting problems:Physics: The Ising and Potts modelsComputer science: counting k-colourings of graphs,counting matchings (monomer-dimer coverings), . . .
![Page 73: Quantum speedup of backtracking and Monte Carlo …csxam/presentations/...Quantum speedup of backtracking and Monte Carlo algorithms Ashley Montanaro School of Mathematics, University](https://reader033.vdocuments.us/reader033/viewer/2022041818/5e5c62c5340a5549f72f9c06/html5/thumbnails/73.jpg)
Application: partition functions
Consider a (classical) physical system which has state space Ω,and a Hamiltonian H : Ω→ R specifying the energy of eachconfiguration x ∈ Ω. Assume that H takes integer values in theset 0, . . . ,n.
We want to compute the partition function
Z(β) =∑x∈Ω
e−βH(x)
for some inverse temperature β.
Encapsulates some interesting problems:Physics: The Ising and Potts modelsComputer science: counting k-colourings of graphs,counting matchings (monomer-dimer coverings), . . .
![Page 74: Quantum speedup of backtracking and Monte Carlo …csxam/presentations/...Quantum speedup of backtracking and Monte Carlo algorithms Ashley Montanaro School of Mathematics, University](https://reader033.vdocuments.us/reader033/viewer/2022041818/5e5c62c5340a5549f72f9c06/html5/thumbnails/74.jpg)
Application: partition functions
Consider a (classical) physical system which has state space Ω,and a Hamiltonian H : Ω→ R specifying the energy of eachconfiguration x ∈ Ω. Assume that H takes integer values in theset 0, . . . ,n.
We want to compute the partition function
Z(β) =∑x∈Ω
e−βH(x)
for some inverse temperature β.
Encapsulates some interesting problems:Physics: The Ising and Potts modelsComputer science: counting k-colourings of graphs,counting matchings (monomer-dimer coverings), . . .
![Page 75: Quantum speedup of backtracking and Monte Carlo …csxam/presentations/...Quantum speedup of backtracking and Monte Carlo algorithms Ashley Montanaro School of Mathematics, University](https://reader033.vdocuments.us/reader033/viewer/2022041818/5e5c62c5340a5549f72f9c06/html5/thumbnails/75.jpg)
Application: partition functions
|Ω| can be exponentially large and Z(β) can be hard tocompute; e.g. #P-hard. So we resort to randomisedmethods for approximating Z(β).
We want to approximate Z(β) up to relative error ε, i.e.output Z such that
|Z − Z(β)| 6 εZ(β).
A standard classical approach: multi-stage Markov chainMonte Carlo (e.g. [Valleau and Card ’72, Stefankovic et al. ’09]).
We can apply the above quantum algorithm to speed upan approximation of expected values in this approach. . .
. . . and we can also replace the classical Markov chainswith quantum walks to get an additional improvement,based on techniques of [Wocjan and Abeyesinghe ’08].
![Page 76: Quantum speedup of backtracking and Monte Carlo …csxam/presentations/...Quantum speedup of backtracking and Monte Carlo algorithms Ashley Montanaro School of Mathematics, University](https://reader033.vdocuments.us/reader033/viewer/2022041818/5e5c62c5340a5549f72f9c06/html5/thumbnails/76.jpg)
Application: partition functions
|Ω| can be exponentially large and Z(β) can be hard tocompute; e.g. #P-hard. So we resort to randomisedmethods for approximating Z(β).
We want to approximate Z(β) up to relative error ε, i.e.output Z such that
|Z − Z(β)| 6 εZ(β).
A standard classical approach: multi-stage Markov chainMonte Carlo (e.g. [Valleau and Card ’72, Stefankovic et al. ’09]).
We can apply the above quantum algorithm to speed upan approximation of expected values in this approach. . .
. . . and we can also replace the classical Markov chainswith quantum walks to get an additional improvement,based on techniques of [Wocjan and Abeyesinghe ’08].
![Page 77: Quantum speedup of backtracking and Monte Carlo …csxam/presentations/...Quantum speedup of backtracking and Monte Carlo algorithms Ashley Montanaro School of Mathematics, University](https://reader033.vdocuments.us/reader033/viewer/2022041818/5e5c62c5340a5549f72f9c06/html5/thumbnails/77.jpg)
Application: partition functions
|Ω| can be exponentially large and Z(β) can be hard tocompute; e.g. #P-hard. So we resort to randomisedmethods for approximating Z(β).
We want to approximate Z(β) up to relative error ε, i.e.output Z such that
|Z − Z(β)| 6 εZ(β).
A standard classical approach: multi-stage Markov chainMonte Carlo (e.g. [Valleau and Card ’72, Stefankovic et al. ’09]).
We can apply the above quantum algorithm to speed upan approximation of expected values in this approach. . .
. . . and we can also replace the classical Markov chainswith quantum walks to get an additional improvement,based on techniques of [Wocjan and Abeyesinghe ’08].
![Page 78: Quantum speedup of backtracking and Monte Carlo …csxam/presentations/...Quantum speedup of backtracking and Monte Carlo algorithms Ashley Montanaro School of Mathematics, University](https://reader033.vdocuments.us/reader033/viewer/2022041818/5e5c62c5340a5549f72f9c06/html5/thumbnails/78.jpg)
Application: partition functions
|Ω| can be exponentially large and Z(β) can be hard tocompute; e.g. #P-hard. So we resort to randomisedmethods for approximating Z(β).
We want to approximate Z(β) up to relative error ε, i.e.output Z such that
|Z − Z(β)| 6 εZ(β).
A standard classical approach: multi-stage Markov chainMonte Carlo (e.g. [Valleau and Card ’72, Stefankovic et al. ’09]).
We can apply the above quantum algorithm to speed upan approximation of expected values in this approach. . .
. . . and we can also replace the classical Markov chainswith quantum walks to get an additional improvement,based on techniques of [Wocjan and Abeyesinghe ’08].
![Page 79: Quantum speedup of backtracking and Monte Carlo …csxam/presentations/...Quantum speedup of backtracking and Monte Carlo algorithms Ashley Montanaro School of Mathematics, University](https://reader033.vdocuments.us/reader033/viewer/2022041818/5e5c62c5340a5549f72f9c06/html5/thumbnails/79.jpg)
Example: The ferromagnetic Ising modelWe are given as input a graph G = (V,E) with n vertices. Weconsider the Ising Hamiltonian
H(z) = −∑
(u,v)∈E
zuzv.
for z ∈ ±1n. We want to approximate
Z(β) =∑
z∈±1n
e−βH(z).
Assume that we have a classical Markov chain whichsamples from the Gibbs distribution in time O(n).This holds for low enough β (depending on the graph G).
Then we have the following speedup:
Best classical runtime known [Stefankovic et al. ’09]: O(n2/ε2)
Quantum runtime: O(n3/2/ε+ n2)
![Page 80: Quantum speedup of backtracking and Monte Carlo …csxam/presentations/...Quantum speedup of backtracking and Monte Carlo algorithms Ashley Montanaro School of Mathematics, University](https://reader033.vdocuments.us/reader033/viewer/2022041818/5e5c62c5340a5549f72f9c06/html5/thumbnails/80.jpg)
Example: The ferromagnetic Ising modelWe are given as input a graph G = (V,E) with n vertices. Weconsider the Ising Hamiltonian
H(z) = −∑
(u,v)∈E
zuzv.
for z ∈ ±1n. We want to approximate
Z(β) =∑
z∈±1n
e−βH(z).
Assume that we have a classical Markov chain whichsamples from the Gibbs distribution in time O(n).
This holds for low enough β (depending on the graph G).
Then we have the following speedup:
Best classical runtime known [Stefankovic et al. ’09]: O(n2/ε2)
Quantum runtime: O(n3/2/ε+ n2)
![Page 81: Quantum speedup of backtracking and Monte Carlo …csxam/presentations/...Quantum speedup of backtracking and Monte Carlo algorithms Ashley Montanaro School of Mathematics, University](https://reader033.vdocuments.us/reader033/viewer/2022041818/5e5c62c5340a5549f72f9c06/html5/thumbnails/81.jpg)
Example: The ferromagnetic Ising modelWe are given as input a graph G = (V,E) with n vertices. Weconsider the Ising Hamiltonian
H(z) = −∑
(u,v)∈E
zuzv.
for z ∈ ±1n. We want to approximate
Z(β) =∑
z∈±1n
e−βH(z).
Assume that we have a classical Markov chain whichsamples from the Gibbs distribution in time O(n).This holds for low enough β (depending on the graph G).
Then we have the following speedup:
Best classical runtime known [Stefankovic et al. ’09]: O(n2/ε2)
Quantum runtime: O(n3/2/ε+ n2)
![Page 82: Quantum speedup of backtracking and Monte Carlo …csxam/presentations/...Quantum speedup of backtracking and Monte Carlo algorithms Ashley Montanaro School of Mathematics, University](https://reader033.vdocuments.us/reader033/viewer/2022041818/5e5c62c5340a5549f72f9c06/html5/thumbnails/82.jpg)
Example: The ferromagnetic Ising modelWe are given as input a graph G = (V,E) with n vertices. Weconsider the Ising Hamiltonian
H(z) = −∑
(u,v)∈E
zuzv.
for z ∈ ±1n. We want to approximate
Z(β) =∑
z∈±1n
e−βH(z).
Assume that we have a classical Markov chain whichsamples from the Gibbs distribution in time O(n).This holds for low enough β (depending on the graph G).
Then we have the following speedup:
Best classical runtime known [Stefankovic et al. ’09]: O(n2/ε2)
Quantum runtime: O(n3/2/ε+ n2)
![Page 83: Quantum speedup of backtracking and Monte Carlo …csxam/presentations/...Quantum speedup of backtracking and Monte Carlo algorithms Ashley Montanaro School of Mathematics, University](https://reader033.vdocuments.us/reader033/viewer/2022041818/5e5c62c5340a5549f72f9c06/html5/thumbnails/83.jpg)
Summary
Quantum computers can speed up two of themost basic tools in classical algorithmics:
Backtracking, for solving constraintsatisfaction problems;Approximating the mean of a randomvariable with bounded variance, forMonte Carlo methods.
In both cases we get a quadratic speedup.
Thanks!
![Page 84: Quantum speedup of backtracking and Monte Carlo …csxam/presentations/...Quantum speedup of backtracking and Monte Carlo algorithms Ashley Montanaro School of Mathematics, University](https://reader033.vdocuments.us/reader033/viewer/2022041818/5e5c62c5340a5549f72f9c06/html5/thumbnails/84.jpg)
Quantum walk in a tree
The quantum walk operates on a T-dimensional Hilbert spacespanned by |r〉 ∪ |x〉 : x ∈ 1, . . . ,T − 1, where r is the root.
The walk starts in the state |r〉 and is based on a set ofdiffusion operators Dx, where Dx acts on the subspace Hxspanned by |x〉 ∪ |y〉 : x→ y:
If x is marked, then Dx is the identity.
If x is not marked, and x 6= r, then Dx = I − 2|ψx〉〈ψx|,where
|ψx〉 ∝ |x〉+∑
y,x→y
|y〉.
Dr = I − 2|ψr〉〈ψr|, where
|ψr〉 ∝ |r〉+√
n∑
y,r→y
|y〉.
![Page 85: Quantum speedup of backtracking and Monte Carlo …csxam/presentations/...Quantum speedup of backtracking and Monte Carlo algorithms Ashley Montanaro School of Mathematics, University](https://reader033.vdocuments.us/reader033/viewer/2022041818/5e5c62c5340a5549f72f9c06/html5/thumbnails/85.jpg)
Quantum walk in a tree
The quantum walk operates on a T-dimensional Hilbert spacespanned by |r〉 ∪ |x〉 : x ∈ 1, . . . ,T − 1, where r is the root.
The walk starts in the state |r〉 and is based on a set ofdiffusion operators Dx, where Dx acts on the subspace Hxspanned by |x〉 ∪ |y〉 : x→ y:
If x is marked, then Dx is the identity.
If x is not marked, and x 6= r, then Dx = I − 2|ψx〉〈ψx|,where
|ψx〉 ∝ |x〉+∑
y,x→y
|y〉.
Dr = I − 2|ψr〉〈ψr|, where
|ψr〉 ∝ |r〉+√
n∑
y,r→y
|y〉.
![Page 86: Quantum speedup of backtracking and Monte Carlo …csxam/presentations/...Quantum speedup of backtracking and Monte Carlo algorithms Ashley Montanaro School of Mathematics, University](https://reader033.vdocuments.us/reader033/viewer/2022041818/5e5c62c5340a5549f72f9c06/html5/thumbnails/86.jpg)
Quantum walk in a tree
The quantum walk operates on a T-dimensional Hilbert spacespanned by |r〉 ∪ |x〉 : x ∈ 1, . . . ,T − 1, where r is the root.
The walk starts in the state |r〉 and is based on a set ofdiffusion operators Dx, where Dx acts on the subspace Hxspanned by |x〉 ∪ |y〉 : x→ y:
If x is marked, then Dx is the identity.
If x is not marked, and x 6= r, then Dx = I − 2|ψx〉〈ψx|,where
|ψx〉 ∝ |x〉+∑
y,x→y
|y〉.
Dr = I − 2|ψr〉〈ψr|, where
|ψr〉 ∝ |r〉+√
n∑
y,r→y
|y〉.
![Page 87: Quantum speedup of backtracking and Monte Carlo …csxam/presentations/...Quantum speedup of backtracking and Monte Carlo algorithms Ashley Montanaro School of Mathematics, University](https://reader033.vdocuments.us/reader033/viewer/2022041818/5e5c62c5340a5549f72f9c06/html5/thumbnails/87.jpg)
Quantum walk in a tree
The quantum walk operates on a T-dimensional Hilbert spacespanned by |r〉 ∪ |x〉 : x ∈ 1, . . . ,T − 1, where r is the root.
The walk starts in the state |r〉 and is based on a set ofdiffusion operators Dx, where Dx acts on the subspace Hxspanned by |x〉 ∪ |y〉 : x→ y:
If x is marked, then Dx is the identity.
If x is not marked, and x 6= r, then Dx = I − 2|ψx〉〈ψx|,where
|ψx〉 ∝ |x〉+∑
y,x→y
|y〉.
Dr = I − 2|ψr〉〈ψr|, where
|ψr〉 ∝ |r〉+√
n∑
y,r→y
|y〉.
![Page 88: Quantum speedup of backtracking and Monte Carlo …csxam/presentations/...Quantum speedup of backtracking and Monte Carlo algorithms Ashley Montanaro School of Mathematics, University](https://reader033.vdocuments.us/reader033/viewer/2022041818/5e5c62c5340a5549f72f9c06/html5/thumbnails/88.jpg)
Quantum walk in a tree
The quantum walk operates on a T-dimensional Hilbert spacespanned by |r〉 ∪ |x〉 : x ∈ 1, . . . ,T − 1, where r is the root.
The walk starts in the state |r〉 and is based on a set ofdiffusion operators Dx, where Dx acts on the subspace Hxspanned by |x〉 ∪ |y〉 : x→ y:
If x is marked, then Dx is the identity.
If x is not marked, and x 6= r, then Dx = I − 2|ψx〉〈ψx|,where
|ψx〉 ∝ |x〉+∑
y,x→y
|y〉.
Dr = I − 2|ψr〉〈ψr|, where
|ψr〉 ∝ |r〉+√
n∑
y,r→y
|y〉.
![Page 89: Quantum speedup of backtracking and Monte Carlo …csxam/presentations/...Quantum speedup of backtracking and Monte Carlo algorithms Ashley Montanaro School of Mathematics, University](https://reader033.vdocuments.us/reader033/viewer/2022041818/5e5c62c5340a5549f72f9c06/html5/thumbnails/89.jpg)
Quantum walk in a tree
Let A and B be the sets of vertices an even and odd distancefrom the root, respectively.
Then a step of the walk consists of applying the operatorRBRA, where RA =
⊕x∈A Dx and RB = |r〉〈r|+
⊕x∈B Dx.
![Page 90: Quantum speedup of backtracking and Monte Carlo …csxam/presentations/...Quantum speedup of backtracking and Monte Carlo algorithms Ashley Montanaro School of Mathematics, University](https://reader033.vdocuments.us/reader033/viewer/2022041818/5e5c62c5340a5549f72f9c06/html5/thumbnails/90.jpg)
Quantum walk in a tree
Let A and B be the sets of vertices an even and odd distancefrom the root, respectively.
Then a step of the walk consists of applying the operatorRBRA, where RA =
⊕x∈A Dx and RB = |r〉〈r|+
⊕x∈B Dx.
![Page 91: Quantum speedup of backtracking and Monte Carlo …csxam/presentations/...Quantum speedup of backtracking and Monte Carlo algorithms Ashley Montanaro School of Mathematics, University](https://reader033.vdocuments.us/reader033/viewer/2022041818/5e5c62c5340a5549f72f9c06/html5/thumbnails/91.jpg)
Quantum walk in a tree
Let A and B be the sets of vertices an even and odd distancefrom the root, respectively.
Then a step of the walk consists of applying the operatorRBRA, where RA =
⊕x∈A Dx and RB = |r〉〈r|+
⊕x∈B Dx.
![Page 92: Quantum speedup of backtracking and Monte Carlo …csxam/presentations/...Quantum speedup of backtracking and Monte Carlo algorithms Ashley Montanaro School of Mathematics, University](https://reader033.vdocuments.us/reader033/viewer/2022041818/5e5c62c5340a5549f72f9c06/html5/thumbnails/92.jpg)
Quantum walk in a tree
Let A and B be the sets of vertices an even and odd distancefrom the root, respectively.
Then a step of the walk consists of applying the operatorRBRA, where RA =
⊕x∈A Dx and RB = |r〉〈r|+
⊕x∈B Dx.
![Page 93: Quantum speedup of backtracking and Monte Carlo …csxam/presentations/...Quantum speedup of backtracking and Monte Carlo algorithms Ashley Montanaro School of Mathematics, University](https://reader033.vdocuments.us/reader033/viewer/2022041818/5e5c62c5340a5549f72f9c06/html5/thumbnails/93.jpg)
Quantum walk in a tree
Let A and B be the sets of vertices an even and odd distancefrom the root, respectively.
Then a step of the walk consists of applying the operatorRBRA, where RA =
⊕x∈A Dx and RB = |r〉〈r|+
⊕x∈B Dx.
![Page 94: Quantum speedup of backtracking and Monte Carlo …csxam/presentations/...Quantum speedup of backtracking and Monte Carlo algorithms Ashley Montanaro School of Mathematics, University](https://reader033.vdocuments.us/reader033/viewer/2022041818/5e5c62c5340a5549f72f9c06/html5/thumbnails/94.jpg)
Applications
There are also a number of combinatorial problems which canbe expressed as partition function problems.
Counting valid k-colourings of a graph G on n vertices:Assume, for example, that the degree of G is at most k/2.
Best classical runtime known: O(n2/ε2)
Quantum runtime: O(n3/2/ε+ n2)
Counting matchings (monomer-dimer coverings) of a graphwith n vertices and m edges:
Best classical runtime known: O(n2m/ε2)
Quantum runtime: O(n3/2m1/2/ε+ n2m)
![Page 95: Quantum speedup of backtracking and Monte Carlo …csxam/presentations/...Quantum speedup of backtracking and Monte Carlo algorithms Ashley Montanaro School of Mathematics, University](https://reader033.vdocuments.us/reader033/viewer/2022041818/5e5c62c5340a5549f72f9c06/html5/thumbnails/95.jpg)
Applications
There are also a number of combinatorial problems which canbe expressed as partition function problems.
Counting valid k-colourings of a graph G on n vertices:Assume, for example, that the degree of G is at most k/2.
Best classical runtime known: O(n2/ε2)
Quantum runtime: O(n3/2/ε+ n2)
Counting matchings (monomer-dimer coverings) of a graphwith n vertices and m edges:
Best classical runtime known: O(n2m/ε2)
Quantum runtime: O(n3/2m1/2/ε+ n2m)
![Page 96: Quantum speedup of backtracking and Monte Carlo …csxam/presentations/...Quantum speedup of backtracking and Monte Carlo algorithms Ashley Montanaro School of Mathematics, University](https://reader033.vdocuments.us/reader033/viewer/2022041818/5e5c62c5340a5549f72f9c06/html5/thumbnails/96.jpg)
Applications
There are also a number of combinatorial problems which canbe expressed as partition function problems.
Counting valid k-colourings of a graph G on n vertices:Assume, for example, that the degree of G is at most k/2.
Best classical runtime known: O(n2/ε2)
Quantum runtime: O(n3/2/ε+ n2)
Counting matchings (monomer-dimer coverings) of a graphwith n vertices and m edges:
Best classical runtime known: O(n2m/ε2)
Quantum runtime: O(n3/2m1/2/ε+ n2m)