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Adiabatic Quantum Optimization Processor Performance
Richard Harris
D-Wave Systems Inc.Burnaby, BC Canada
September 2009
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Outline
1 C1 Ising Spin Processor
2 AQC Processor Performance
3 Problem Dissection
4 AQO Processor Performance
5 Is This A Quantum Processor?
6 Conclusions and Whats Next
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C1 Ising Spin Processor
Outline
1 C1 Ising Spin Processor
2 AQC Processor Performance
3 Problem Dissection
4 AQO Processor Performance
5 Is This A Quantum Processor?
6 Conclusions and Whats Next
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C1 Ising Spin Processor
Optimization Problem Specification
Let there be an optimization problem of the form
Ec(s) = i
hisi +i,j>i
Ksisj (1)
where
1
hi
+1 and
1
K
+1, for which there exists an
optimal solution s that minimizes the energy Ec(s), where the individualelements si = 1. This problem can be mapped onto a dimensionless Isingspin Hamiltonian of the form
H(t) =
i
hi(i)z +
i,j
Ki,j(i)z
(j)z (2)
Here, the groundstate configuration of the spins s represents the lowestenergy solution to the optimization problem. The objective is toimplement an algorithm in hardware that reliably finds this groundstate.
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C1 I i S i P
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C1 Ising Spin Processor
Adiabatic Quantum Optimization (AQO)
Introduce quantum mechanical degrees of freedom to create a quantumIsing spin Hamiltonian of the form
H(t) =
i
hi(i)z +
i,j
Ki,j(i)z
(j)z
i
f(t)(i)x (3)
Use physical system to find the groundstate by evolving f(t) such that
f(0) hi,Kf(t ) hi,Ki,j
See Farhi et al, Science 292, 472 (2001).
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C1 I i S i P
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C1 Ising Spin Processor
Implementation
CJJ rf-SQUID Flux Qubits
Low energy rf-SQUID Hamiltonian can be approximated as a qubit:
Hq = 12
[z + qx] (4)
where = 2I
pq
xq is the bias energy and q is the tunneling energy. Assuch,
Ipq and q are the defining properties of a flux qubit.
We use compound-compound Josephson junction (CCJJ) rf-SQUID flux
qubits. Here, an external flux bias
x
ccjj controls the tunnel barrier height,thus allowing us to initialize qubits with q kBT and then annealingthe qubit by slowly raising the tunnel barrier such that q 0. A fulldisclosure of the CCJJ rf-SQUID flux qubit has been posted asarXiv:0909.4321.
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C1 Ising Spin Processor
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C1 Ising Spin Processor
Implementation
Example Ipq and
0.67 0.665 0.66 0.655 0.65 0.6450.4
0.6
0.8
1
1.2
1.4
xccjj/0
|Ip q|(A)
a) DataTheory
0.67 0.665 0.66 0.655 0.65 0.64510
5
106
107
108
109
1010
xccjj /0
q
/h(Hz)
b)kbT/h
MRT Data
LZ Data
g| Ipq |g Data
Theory
Theory from CCJJ rf-SQUID Hamiltonian with independentlycalibratedparameters. MRT (incoherent regime), see Harris et al, PRL 101, 117003(2008). LZ (incoherent regime), see Johannson et al, PRB 80, 012507
(2009). g| Ipq |g (coherent regime), see Harris et al, arXiv:0909.4321.
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C1 Ising Spin Processor
Implementation
Flux Qubits are Used in the Flux Basis in AQO
Qubits are used in the flux basis in AQO, not the energy eigenbasis as inmany (but not all) implementations of gate model quantum computation.
5 2.5 0 2.5 5
0
5
10
15
20
25
(q 0/2)/Lq (A)
Energy/h(GH
z)
||
|g
|e
State Energy Basis Flux Basis
|0 |g | |1 |e |
Groundstate Eigenstate, |0 (|0 + |1) /2Excited State Eigenstate, |1 (|0 |1) /2
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C1 Ising Spin Processor
Implementation
A Network of Inductively Coupled rf-SQUID Flux Qubits
Use rf-SQUID flux qubits as quantum Ising spins. Provide tunable mutualinductances between qubits to implement pairwise interactions. Lowenergy Hamiltonian for general bias conditions can be written as
H(t) = 1
2i
x
i (t)2|Ip
i
x
ccjj(t) |
(i)
z + q
x
ccjj(t)
(i)
x
+i,j
Mi,j
cpli,j
Ipq
xccjj(t)2 (i)z (j)z (5)
x
i ,x
ccjj qubit external flux biasesMi,j(
cpli,j ) tunable mutual inductance between qubitsq qubit tunnel energy|Ipq| qubit persistent current
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C1 Ising Spin Processor
Implementation
Mapping Problems onto Hardware
Assume that all qubits are identical: |Ipi (t)| |Ipq(t)| and i(t) (t).Map problem parameters hi and K in the dimensionless problemHamiltonian onto the hardware Hamiltonian. Scale problem energies tothe antiferromagnetic (AFM) coupling energy.
H(t) = JAFM(t)
i
hi(i)z +
i,j
Ki,j(i)z
(j)z
1
2
i
(t)(i)x (6)
hi JAFM(t) = |Ipq(t)|xi (t)K JAFM(t) = M|Ipq(t)|2
JAFM(t) = MAFM|Ipq(t)|2
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g p
Implementation
Removing Time-Dependence In Problem Mapping
The problem parameters hi and K are supposed to be time-independent.However, the mapping onto physical hardware yielded
hi =|Ipq(t)|qi (t)
MAFM|Ipq(t)
|2
=xi (t)
MAFM|Ipq(t)
|(7)
K =M|Ipq(t)|2
MAFM|Ipq(t)|2 =M
MAFM(8)
Clearly K are naturally time-independent. However, in order to render hi
time-independent one must apply time-dependent qubit flux biases
xi (t) = hi MAFM|Ipq(t)| (9)
See Harris et al, arXiv:0903.3906.
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g p
Implementation
Running the AQO Algorithm
To run AQO on this hardware, use given mappings for hi and K and sweepqubit tuning parameters from regime where JAFM to JAFM.
H(t) = JAFM(t)
i
hi(i)z +
i,j
Ki,j(i)z
(j)z
12
i
(t)(i)x (10)
0.63 0.64 0.65 0.66 0.670
0.05
0.1
0.15
0.2
xcjj/0
Temperature(K)
T
/2kBJAFM/kB
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g
C1 Processor Architecture
Processor Unit Cell
Hardware has been detailed in several publications. Key references:
Programmable Control Circuitry: Johnson et al, arXiv:0907.3757
High Fidelity Readout: Berkley et al, arXiv:0905.0891
Inter-qubit Coupler: Harris et al, PRB 80, 052506 (2009)
CCJJ Qubit: Harris et al, arXiv:0909.4321
A new publication describing how this hardware is operated is currently
being drafted and will be posted on arXiv soon.
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C1 Processor Architecture
Processor Unit Cell
8 flux qubits (shaded) connected by 16 inter-qubit couplers (dark).
q0 q1 q2 q3
q4
q5
q6
q7RO
CCJJLT
CO
IPC
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Embedding Problems on C1 Processor
Optimization Problems As Graphs
Problems posed as connected graphs. One can assign 1 hi +1 toany vertex and 1 K +1 to any edge.
0
12
3
4
5 6
7
+1
1
0
Qubit
Coupler
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Embedding Problems on C1 Processor
Optimization Problems on Hardware
Assign vertices and edges of graphs (problems) to qubits and couplers,respectively. Qubits are extended objects in the hardware and couplers arelocalized at intersections of qubits.
0 1 2 3
4
5
6
7
+1
1
0
Qubit
Coupler
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Embedding Problems on C1 Processor
Symmetry Leads to Multiple Embeddings
There are many ways to embed the same graph. Some examples . . .
0 1 2 3
4
5
6
7
Reference
0 1 2 3
4
5
6
7
/2 Rotation
0 1 2 3
4
5
6
7
Rotation
0 1 2 3
4
5
6
7
3/2 Rotation
0 1 2 3
4
5
6
7
Reflect Across x=0
0 1 2 3
4
5
6
7
Reflect Across y=0
0 1 2 3
4
5
6
7
Reflect Across y=-x
0 1 2 3
4
5
6
7
Reflect Across y=x
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AQC Processor Performance
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Outline
1 C1 Ising Spin Processor
2 AQC Processor Performance
3 Problem Dissection
4 AQO Processor Performance
5
Is This A Quantum Processor?
6 Conclusions and Whats Next
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AQC Processor Performance
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Benchmarking The Hardware
Definition of Benchmark Problem Set
A set of 8 permutations of 800 unique graphs (total 6400 problems) weregenerated. The graphs were created by choosing sets of 8 values of hi and16 values of K from a uniform random distribution of values within the set
hi,K [ 1 6/7 5/7 . . . +5/7 +6/7 +1 ].
We denote such a problem as being specified to 4 bits of precision (4BOP),where hi,K are sampled from 2
4 1 evenly spaced values between 1.
Connections To Complexity and Noise
The precision to which one wants to specify values of hi and K is relatedto the complexity or difficulty of the optimization problem that is to besolved. The precision to which one can specify values of hi and K inpractice is intimately linked to the physics behind dephasing via flux noisein superconducting gate model quantum computing hardware.
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AQC Processor Performance
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Control Signals
Only 2 time-dependent control signals during annealing: xccjj(t) and asingle persistent current compensation signal Ig(t) that simultaneously
drives all qubit flux biases xi (t).
0
0
0
-0/2
-0/2
~0/3
iro(t)
ccjj(t)x
ro(t)x
xlatch(t)
t=0 trep
QFPs
}d
c-SQUID
-0
-0
Ig(t)
100 s
}}
Qubits
ReadoutAnneal
ccjjm
ccjjb
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AQC Processor Performance
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AQC Processor Performance
Adiabatic quantum computation (AQC) implies system should be in
groundstate at all times with P = 1. Run each of the 6400 problems 1024times and bin the probability of finding the groundstate. Data showstructure around P 0.6 and a tail that is rapidly suppressed at lower P.
0 0.2 0.4 0.6 0.8 10
200
400
600
800
1000
Probability
Frequency
8Q4BOP, Probability of Correct SolutionW444.15.R3C13
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AQC Processor Performance
f QC
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Failures of the AQC Processor
Track the probability of finding the system in the most prominent incorrect
state. If the errors were simply random, P should be very low for all cases.Data indicate otherwise.
0 0.2 0.4 0.6 0.8 10
200
400
600
800
1000
1200
1400
Probability
Frequency
8Q4BOP, Probability of Most Probable Incorrect AnswerW444.15.R3C13
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AQC Processor Performance
S I U Th f
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Summing It Up Thusfar ...
The Key Experimental Observations
The hardware seemingly only yields a groundstate with P = 1 for only aportion of the benchmarking problems. However, the failures do not
appear to be random.
Should one even expect this system to return a groundstate with P = 1?
The remainder of this lecture will address the question posed above. It willbe demonstrated that acknowledging that the AQO processor is operated
at finite temperature is essential to understanding its performance.
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Problem Dissection
O li
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Outline
1 C1 Ising Spin Processor
2 AQC Processor Performance
3 Problem Dissection
4 AQO Processor Performance
5 Is This A Quantum Processor?
6 Conclusions and Whats Next
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Problem Dissection
A T i l P bl
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A Typical Problem
Consider Problem 793 (permutation 0 pictured below). Graph contains a
seemingly typical mix of hi and K, nothing special in its structure. Thiswas a problem that returned the groundstate with P 0.6 0.1 for allpermutations.
0
12
3
4
5 6
7
+1
1
0
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Problem Dissection
St t Ab th Si l C t L l
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States Above the Single Count Level
Permutation State (
i
si+1
2 28i
) State (s) Probability0 200 (1 1 -1 -1 1 -1 -1 -1) 563/1024 *0 201 (1 1 -1 -1 1 -1 -1 1) 91/10240 232 (1 1 1 -1 1 -1 -1 -1) 362/1024
. . . .
. . . .
. . . .
7 140 (1 -1 -1 -1 1 1 -1 -1) 656/1024 *7 142 (1 -1 -1 -1 1 1 1 -1) 212/10247 156 (1 -1 -1 1 1 1 -1 -1) 152/1024
Key Observation
Only 3 states showed up for all 8 embeddings of the same problem. Theerror states are always related to the groundstate (*) by a single bit flip.
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Problem Dissection
A P tt I th E St t s
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A Pattern In the Error States
Permutation Bits That Fail Mapping Transformed Bits
0 2,7 Reference 2,71 0,6 Rotated +pi/2 2,72 1,4 Rotated pi 2,73 3,5 Rotated -pi/2 2,7
4 1,7 Reflect across V 2,75 2,4 Reflect across H 2,76 0,5 Reflect across x=-y 2,77 3,6 Reflect across x=+y 2,7
Key Conclusion
The failures are independent of permutation. The loss of probability in thegroundstate is due to something intrinsic to the problem structure.
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Problem Dissection
The Failure States are Low Lying States
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The Failure States are Low Lying States
Choose permutation 0 and calculate the energy for all classical states s:
Ec(s) = JAFM
7i=0
hisi +7
i,j>i
Ksisj
(11)
0 50 100 150 200 2500
5
10
15
20
State (dec)
(EE
g)/JAFM
8Q4BOP, Problem 793 Classical Energy Versus State
State 200 (Groundstate) State 201
State 232
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Problem Dissection
Temperature Matters
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Temperature Matters
Thermalization of the Processor State
If our hardware really does implement an internally adiabatic evolutioncoupled to a bath at temperature T, then one really ought to observe athermal distribution of processor states si:
Pi =
eEc(si)/kBT
Z (12)Z=
i
eEc(si)/kBT
For the C1 chip, we know T
50mK and MAFM = 1.563 pH. Since JAFM
grows during annealing, one can speculate that the device will successfullythermalize until it enters the incoherent regime where the dynamics will befrozen: the dividing line is roughly located at xcjj 0.66 0, at whichpoint
Ipq
1.1A. This yields JAFM/kB = MAFM
Ipq
2/kB 150 mK.
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Problem Dissection
Temperature Matters
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Temperature Matters
A Quick Sanity Check
Compare observed (averaged over 8 permutations) and predictedprobabilities for problem 793:
State P Observed P Predicted
200 0.55 0.58232 0.35 0.29201 0.09 0.12
Others 0.01 0.01The thermalization picture does pick out the right states and gives adecent estimate of their probabilities.
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Problem Dissection
Annealing Trajectories and Thermalization
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Annealing Trajectories and Thermalization
Consider lowest 16 levels from a simulation of the system running Problem
793: System localizes into classical configurations when x
ccjj 0.65 0(where q 2JAFM). We see evidence of the device thermalizing up toxccjj 0.66 0, which is well beyond the phase transition.
0.635 0.64 0.645 0.65 0.655 0.66 0.665 0.670
0.1
0.2
0.3
0.4
0.5
0.6
0.7
xccjj/0
(E
E0)/kB
(K)
Coherent Incoherent
|20 0
|23 2
|20 1
|0
|1,m
|2,m,n
3T
Second Excited Stateof rfSQUID
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Problem Dissection
Is This a 1-Qubit Fluke?
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Is This a 1-Qubit Fluke?
The low lying states for problem 793 were both 1 bit flip from the
groundstate. Is there evidence of low lying states that are multiple bit flipsfrom the groundstate being thermally occupied? Consider Problem 698:
0
12
3
4
5 6
7
+1
1
0
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Problem Dissection
Experimental (Theoretical) Occupation of Ground State
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Experimental (Theoretical) Occupation of Ground State
0 50 100 150 200 2500
2
4
6
8
10
12
14
State (dec)
(EE0
)/JAFM
8Q4BOP, Problem 698 Classical Energy Versus State
Groundstate: 0.36 (0.37)
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Problem Dissection
Experimental (Theoretical) Occupation of Excited States
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Experimental (Theoretical) Occupation of Excited States
0 50 100 150 200 2500
2
4
6
8
10
12
14
State (dec)
(EE0
)/JAFM
8Q4BOP, Problem 698 Classical Energy Versus State
1st Excited: 0.16 (0.17) 1st Excited: 0.20 (0.17)
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Problem Dissection
Experimental (Theoretical) Occupation of Excited States
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Experimental (Theoretical) Occupation of Excited States
0 50 100 150 200 2500
2
4
6
8
10
12
14
State (dec)
(EE0
)/JA
FM
8Q4BOP, Problem 698 Classical Energy Versus State
2nd Excited: 0.04 (0.08) 2nd Excited: 0.06 (0.08)
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Problem Dissection
Experimental (Theoretical) Occupation of Excited States
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Experimental (Theoretical) Occupation of Excited States
0 50 100 150 200 2500
2
4
6
8
10
12
14
State (dec)
(EE0
)/JA
FM
8Q4BOP, Problem 698 Classical Energy Versus State
3rd Excited: 0.02 (0.04)3rd Excited: 0.07 (0.04)
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Problem Dissection
Experimental (Theoretical) Occupation of Excited States
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p ( ) O p S
0 50 100 150 200 2500
2
4
6
8
10
12
14
State (dec)
(EE0
)/JA
FM
8Q4BOP, Problem 698 Classical Energy Versus State
4th Excited: 0.02 (0.02) 4th Excited: 0.02 (0.02)
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Problem Dissection
Experimental (Theoretical) Occupation of Excited States
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p ( ) p
0 50 100 150 200 2500
2
4
6
8
10
12
14
State (dec)
(EE0
)/JA
FM
8Q4BOP, Problem 698 Classical Energy Versus State
5th Excited: 0.01 (0.01)
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Problem Dissection
Observation of Multiple Bit Flips
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p p
State Hamming Distance P Observed P Predicted
97 - 0.36 0.37101 1 0.20 0.1754 5 0.16 0.1784 4 0.06 0.0820 5 0.04 0.08
117 2 0.07 0.04190 7 0.02 0.0455 4 0.02 0.0239 4 0.02 0.02
227 2 0.01 0.01
Others - 0.01 0.01
Multi-bit differences are observed with appreciable probability. Thediscrepancies between experiment and theory may be due to the target
Hamiltonian being slightly miscalibrated.Copyright, D-Wave Systems (2009) AQO Processor Performance September 2009 39 / 59
Problem Dissection
Simulated Processor Performance
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Nature Wont Allow a Perfect AQC ...
The humped structure appears to be due to a conspiracy betweenthermalization and problem set structure - it will never go away at finite T.
0 0.5 10
200
400
600
800
1000
Probability
Frequency
8Q4BOP, Probability of Correct SolutionW444.15.R3C13
0 0.5 10
200
400
600
800
1000
Probability
Frequency
8Q4BOP, Probability of Correct SolutionTHERMAL SIMULATION
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Problem Dissection
Simulated Processor Performance
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Nature Wont Allow a Perfect AQC ...
Adding a small amount of Gaussian distributed random noise on hi to thesimulation reproduces the salient features.
0 0.5 10
200
400
600
800
1000
Probability
Frequency
8Q4BOP, Probability of Correct SolutionW444.15.R3C13
0 0.5 10
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8Q4BOP, Probability of Correct SolutionSIMULATION, Gaussian Randomization
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AQO Processor Performance
Outline
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1 C1 Ising Spin Processor
2 AQC Processor Performance
3 Problem Dissection
4 AQO Processor Performance
5 Is This A Quantum Processor?
6 Conclusions and Whats Next
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AQO Processor Performance
Choosing a New Performance Metric
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AQO = AQC when T > 0Simply counting the probability of achieving the groundstate alone candistort ones perception of processor performance. When characterizingAQO, one really ought to specify the probability of obtaining a low energystate within a thermal energy window above the groundstate.
So what is a fair width E for this window? Consider a system that hastwo low lying levels that are well separated from all others. Let us definethe success metric as being a probability of 0.95 of being within thegroundstate. Solving for the excited state probability then yields:
0.05 =eE/kBT
1 + eE/kBT
E 3kBT
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AQO Processor Performance
A Dividing Line Between Low and High Energy States
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Revisit the classical energy plot for problem 793, permutation 0: The
suggested definition of E = 3kBT makes sense.
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(EEg
)/JAFM
8Q4BOP, Problem 793 Classical Energy Versus State
EState 200 (Groundstate) State 201
State 232
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AQO Processor Performance
AQO Performance
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Low Energy Solutions
The hardware returns low energy solutions with very high probability. Thisis the expected behaviour of a AQO processor operated at finite T.
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8Q4BOP, Solution Within Thermal BandW444.15.R3C13
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8Q4BOP, Solution Within Thermal BandTHERMAL SIMULATION
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AQO Processor Performance
Number of Low Lying States
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How many states lie within the thermal window?
The number is 1, which seems to be a feature ofthe choice of test problems. This means that finding the true groundstateout of relatively few occupied states is easy.
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requency
8Q4BOP, Number of States Within Thermal BandW444.15.R3C13
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AQO Processor Performance
A Scalar Success Metric
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Is the groundstate the most probable state?Thermodynamics dictates that the groundstate ought to be the mostprobable state. We ran each problem 1024 times, identified the state sthat showed up with the highest probability and then asked whether thisstate was a groundstate: This was confirmed to be the case for 6073 out
of 6400 problems (success rate 95.89%).
Even for those cases where a groundstate was not the most probable state,a groundstate was always observed with finite probability. Since only a
small number of states ( 28
) showed up with finite probability for allproblems in the benchmarking set, then it was very easy to identify anexact solution to a given optimization problem.
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AQO Processor Performance
Implications for Hardware Operation
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Differing Modes of Hardware OperationSuggested modes for operation of the AQO processor:
Run problem a statistically large number of times, calculate
Ec(s) =
i
hisi +
i,j
Ki,jsisj
for all states s that appear above the readout noise level. Take statethat yields the lowest Ec as the solution.
Run hardware once, calculate Ec using measured state s. If Ec is less
than some threshold that is dictated by the user, then answer can bedeemed an acceptable solution. If Ec > threshold, iterate.
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AQO Processor Performance
Number of Repetitions
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How many times must one run the hardware to find an exact solution?
Assume for some particular problem that one obtains the groundstate withprobability 0.05, so 95% of the time one gets another state. In order toobserve the groundstate at least once in M measurements with probabilityPgs,
1
Pgs = (0.95)
M (13)
Pgs M
0.95 590.99 90
0.999 135. . . . . .
Even in the worst case scenario for this processor, one could find an exactsolution to an optimization problem in a small number of iterations.
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Is This A Quantum Processor?
Outline
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1 C1 Ising Spin Processor
2 AQC Processor Performance
3
Problem Dissection
4 AQO Processor Performance
5 Is This A Quantum Processor?
6 Conclusions and Whats Next
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Is This A Quantum Processor?
Is This a Quantum Processor?
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Point 1:
The data shown herein have simply demonstrated that we have builthardware that solves optimization problems in a manner that is consistent
with thermodynamics. As such, they cannot be used to argue whether thisis a classical or quantum mechanical system as all physical systems mustobey thermodynamics. Rather, the details must be in how the systemthermalizes.
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Is This A Quantum Processor?
Is This a Quantum Processor?
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Point 2:
The fact that measured single qubit parametersIpq and agree with the
results from an independently calibrated quantum mechanical rf-SQUID
Hamiltonian indicates that the constituent elements of this processor arequantum mechanical. However, this does not immediately imply quantummechanical behavior of the collective system. We need to perform specific,targeted experiments to characterize the dynamics of the collective systemand compare the results to quantitative models, be they quantum or
classical, of the system evolution.
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Is This A Quantum Processor?
... But, System Thermalizes in the Presence of Large Tunnel Barriers
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The system thermalizes when the single qubit tunnel barrier heightU kBT, but only in the presence of finite tunneling energy . We see
signatures of thermalization up to x
cjj 0.66 0, where U/kBT 10but /h 50 MHz. Thermal activation over such a barrier seems unlikely,but tunneling is still relatively fast.
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0.3
0.4
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xcjj/0
T
emperature(K)
T
/2kB
JAFM/kB
U/kB
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Conclusions and Whats Next
Outline
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1 C1 Ising Spin Processor
2 AQC Processor Performance
3 Problem Dissection
4 AQO Processor Performance
5 Is This A Quantum Processor?
6 Conclusions and Whats Next
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Conclusions and Whats Next
Conclusions
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It is critical to include the effects of finite T when discussing theperformance of a AQO processor.
The dominant failure mode for the AQO processor can be describedby thermalization processes. These are arguably soft failures to lowenergy solutions. The width of the band of probable low lying statesis given roughly by 3kBT.
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Conclusions and Whats Next
Whats Next - The Key Issues
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Two Key Issues That Need to Be Addressed:
Scaling: 8-qubit system is too small to state anything meaningfulregarding the scaling of AQO. We need to experimentally characterizelarger processors at a size that at least pushes the performance of
classical optimization systems (hardware and software).Mechanism: We need to perform carefully crafted experiments toconfirm or refute that quantum mechanics is essential to theprocessor operation. This boils down to determining whether theoff-diagonal elements of the single qubit Hamiltonians (q) affect thedynamics of the collective processor.
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Conclusions and Whats Next
Whats Next - Scaling
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We currently have three 128-qubit chips cold and being calibrated at themoment. Our intention is to benchmark these devices against large sets ofrandom Ising spin glass problems.
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Conclusions and Whats Next
Whats Next - Mechanism
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We are looking into using antiferromagentic chains of varying length toprobe the dynamics of collective systems. Such chains have 2 low lyingstates that are well separated from all others, thus simplifying modelling.
Meff
1Q:F2
x
F1x
F2x
F2x
F3x=0
F3x=0 F4
x=0
MeffMeffMeff
MeffMeff
F1x
2Q:
3Q:
A
F1x
0
0
-0/2
-0/2
~0/3
i t
ccjj(t)x
ro(t)x
xlatch(t)
QFPs
dc-S
QUID
-0
-0
Ig(t)
}}
Qubits
ccjjm
ccjjb
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Conclusions and Whats Next
Whats Next - Mechanism
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By carefully choosing static flux biases applied to the antiferromagnetic
chain one can steer the magnitude of the gap at the anticrossing0 < g < q. This provides a means for slowing down the dynamics of thecollective system such that it can be probed with our low bandwidth biaslines.
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E)/h(GHz)
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