Entanglement

This word was never used before 1994. Meanwhile physicists understood:

* The structure of atoms, molecules, and nuclei

* Quantum electrodynamics

* Solid state physics

* NMR

* Superconductivity

* Lasers

* Quantum optics

* Quantum Hall Effect

........

Is this really a useful notion?

Can we have and do we need quantum computation?

Michel Dyakonov

Université Montpellier II, France

Outline:* Overview by an outsider

Brief history and general ideas Quantum computing with... Error correction and threshold theorem

* Relaxation (decoherence)

* Fault-tolerant quantum computation

* Challenge

* The lessons from building perpetual motion machines

* Much more powerful in doing what ?

* Conclusions

Is fault-tolerant quantum computation really possible?M.I. Dyakonov, 2007, arxiv.org/abs/quant-ph/0610117

Quantum computing: a view from the enemy campM.I. Dyakonov, 2001, arxiv.org/abs/cond-mat/0110326

Quantum Computation: A Grand

Mathematical Challenge for the Twenty-

First Century and the Millennium

S. J. Lomonaco, Jr., ed. American Mathematical Society, Providence, Rhode Island (2002)

A grand challenge for the Millennium ...

Philosopher's Stone (Lapis Philosophorum):

A Grand Challenge for the Eleventh Century

and the Millennium

S. J. Abrucci, Jr., ed. Roman Philosophical Society, Pisa (1002)

Brief history

R. Feynman (1982): Nature isn’t classical, dammit, and if you want to make a simulation of Nature, you’d better make it quantum mechanical, and by golly it’s

a wonderful problem, because it doesn’t look so easy

D. Deutsch (1985): Universal quantum computer

P. Shor (1994): Factoring by quantum computing (Shor’s algorithm)

P. Shor (1995), A. Steane (1996): Quantum error correction by encoding

P. Shor (1996) and others: Fault-tolerant quantum computation (technical instruction)

D. Aharonov and M. Ben-Or (1999): The “threshold theorem”

L. Vandersypen et al (2001): 15 = 3 × 5 by NMR (experimental)

The general idea of quantum computing

Quantum computing is supposed to use some 103 – 106 two-level systems (spins), called “QUBITS ”

Each spin can be in a state a|↑ + b|↓ or a|0 + b|1

The state of the whole computer is described by the grand wave function

Ψ = A0|000...00 + A1|000...01 + A2|000...10 + ... + A 2N

–1 |111...11

with 2N complex amplitudes Ak

NB: 21000 ~ 10300 is much greater than the number of protons in the Universe ( ~1080 )

The computing process consists of manipulating ALL of these continuously changing

variables and measuring all qubits at the end

An elementary unitary transformation of amplitudes is called “QUANTUM GATE ”

IF there is no noise (spin relaxation), IF the gates are ideal, and IF measurements are perfect,

THEN it is proven that after applying ~1010 gates, one can use Shor's algorithm to factorize

numbers like ~ 10130 faster than any classical computer

* neutral atoms * buckyballs

* trapped ions, atoms, and light * spin qubits interacting

* neutral atoms in an optical lattice through delocalized excitons

* quantum dots * non-deterministic gates

* globally controlled interactions * optically hole burnt materials

* quantum dots in a microcavity * global one-and two-qubit gates

* endohedral fullerenes * magnetic atoms in optical

* “always on” Heisenberg interaction lattices of reduced periodicity

* spatially delocalized qubits * superconductors

* Hydrogenic spin in Silicon * collective ensembles of multi-

* atomic Josephson junction arrays level systems

* rare-earth-ion doped crystals * hyperfine clock states

* magnetically interacting atoms * small-amplitude coherent states

* perpetually coupled qubits * superconducting circuits

* only one mobile quasiparticle * trapped electron states

* "enhancement" quantum dots * highly verified cluster states

* polarization encoding * toric code states

* phase modulated laser pulses * bosonic atoms

Quantum computing with (randomely picked proposals):

* superconducting charge qubits * slow measurements

* continuous-variable cluster states * scattering matrices

* Kerr-nonlinear photonic crystals * qudits

* polar molecules * unlabeled qubits

* transverse phonon modes * shutter logic

* abelian anyons on the honeycomb lattice * matrix product states

* v = 5/2 fractional quantum Hall state * probabilistic quantum gates

* diatomic bits in optical lattices * perpetually coupled spins

* semiconductor nanostructures * cold bosonic atoms

* atoms in periodic potentials * para-hydrogen

* probabilistic ion-photon mapping * ideal Clifford gates

* optical coherent states * cluster states

* one-dimensional optical lattice * advice

* un-tunable couplings * unknown parameters

* vibrationally excited molecules * superconducting circuits

* squeezed coherent states * Heisenberg ABAB chain

* spin-1/2 pairs * harmonic oscillators

* ballistic electrons * cavity QED systems

* mesoscopic superposition states * abelian anyons

Quantum computing with (continued):

Topological quantum computing with anyons

(ν = 5/2 fractional quantum Hall plateau)

Hence the obvious proposal: to use these hypothetical anyons for quantum computing. We must move the anyons around one another making complex topological structures, so that the knot theory may be used.

The observed ν = 5/2 fractional quantum Hall plateau does not fit into the composite fermion concept. Some people think that this is a manifestation of anyons. Others think differently. Nobody knows.

This

Schematic presentation of the Hall bar (3×6 mm),containing 1010 2D electrons (and presumably an equal number of anyons)

See also « Anyon theory of high Tc superconductivity »

Quantum computing with...

Google gives 55 600 hits for « quantum computing with »

and 75 900 hits for « quantum computation with »

Isn't this crazy?... Is there anything at all that is NOT suitable for quantum computing?

Some possibilities are still not explored:

Quantum computing with:

* muons (muon qubits for fault-tolerant quantum computing)

* neutrinos (very good candidate, because of long decoherence time)

* quarks (easily produced at high energy facilities. Advantage: one can use flavor and SU(3) symmetry)

* rotating black holes (can be used as quantum gates for neutrinos)

* Shrödinger cats (entanglement easily observed at room temperature)

15 = 3 × 5 (experimental application of the Shor algorithm with NMR)

L. Vandersypen et al (2001)

Ideally expected spectrum →

Experimental spectrum →

Simulation of decoherence effects →

NMR spectrum demonstrating factorization of 15 by Shor

Postulates of Quantum Mechanics

Ψ = A|0 + B|1

Measure the state of a qubit. The result is (0) or (1) with probabilities |A|2 and |B|2

Measurement gives (0) → the wavefunction collapses to the state Ψ = |0

Basic principle of error correction

Moreover:

Consider our spin + the rest of the world (ROTW) !

The wavefunction can be written as:

Ψ = A |0 0 + B |1 1 ,

where 0 and 1 are some two states of the ROTW

We now measure the state of our qubit only. Depending on whether the result is (0) or (1),

not only the wavefunction of our qubit will collapse to |0 or |1, but also the wavefunction of the

ROTW will collapse to 0 or 1 respectively!

Ψ = A|0 + B|1

Logical qubits are encoded by special superpositions of physical qubits

Error correction by encoding

Ancilla qubits, syndrome extraction operator, measure ancilla, correct error

Fault-tolerant quantum computation – a technical instruction, P. Shor (1996)

In fact, concatenated encoding should be used, in which each qubit on the right will be in its turn encoded in the same manner, and so on...

A. Steane (1996)

Concatenated encoding

The future quantum engineer will have to encode one logical qubit

by maybe 73=343 qubits...

(J. Preskill, 1997)

Very old Russian joke

Pushing the green button annihilates the enemy ground forces

White button: same for air force

Blue button: same for the navy

It's up to the engineers to work out the details

View of the Supreme Commander work-desk

The threshold theorem

Concatenated encoding Parallel error correction for different qubits Noise model: uncorrelated in space and time errors in individual qubits,

gates, and measurements

Once the error rate per qubit per gate is below a certain value ε,

indefinitely long quantum computation becomes feasible, even if all of

the qubits involved are subject to relaxation processes, and all the

manipulations with qubits are not exact.

D. Aharonov and M. Ben-Or (1999)

Estimate for the threshold: ε ~ 10–6 – 10–4

Beware theorems

Dream...

The mathematical method for capturing a Lion in the desert

The desert D being a separable topological space, it contains a countable subset S that is everywhere dense therein. (For example, the set of points with rational coordinates is eligible as S.)

Therefore, letting be the point at which the Lion is located, we can find a sequence

, with limn→ ∞ xn = x

This done, we approach the point x along the

sequence {xn} and capture the Lion

Snnx

Dx

Beware theorems

The real thing...

... and reality

The Lion, by Rembrandt

Relaxation (decoherence)

Mystification of spin relaxation in QC literature:

“The qubit (spin) gets entangled with the environment…”

“The environment is constantly trying to look at the state of a qubit, a process called decoherence”

* * * * * * * * * * *

Meanwhile relaxation of 2-level systems was studied a lot and is quite well understood

Spin relaxation is a result of the action of fluctuating in time magnetic fields

A randomly fluctuating magnetic field is characterized by its correlation time, c,

and by the average angle of spin precession, α, during time c

For the most frequent case when α << 1 , the spin vector experiences a slow angular diffusion

RMS angle after time t >> c is ε = α(t/c)1/2

Then the relaxation time is = c α2

Relaxation (decoherence)

This process goes on continuously for all the qubits within the quantum computer

Each qubit loses memory of its initial state during the relaxation time τ

Thus, at the final stage of the Shor algorithm, supposing that you managed to achieve a

good wavefunction (of 73 × 1000 qubits, encoding 1000 logical qubits), the states of all

343000 qubits should be measured during a time considerably shorter than τ/343000

Surprisingly, nobody has ever discussed this problem

(it is for the future quantum engineers to figure this out)

Important point:

The wavefunction of N qubits deteriorates faster than that,

during a characteristic time τ/N

(correlations are more fragile, they disappear faster)

Quantum computing with “decoherence-free subspaces”

IDEA: Suppose that, while individual qubits experience decoherence, there exist (due to some symmetry) a subset of many-qubit states that do not relax.

We can then encode information within this decoherence-free subspaceand safely perform fault-tolerant quantum computations of arbitrary length!

(A large and respectable branch of mathematics, considering which is the best way to do this, see for example Theorem 4 below)

Unfortunately, decoherence-free subspaces do not exist,

except in some imaginary world

How to factorize numbers by quantum computing?

To beat my PC in factorizing numbers, one must have at least 1000 qubits

(without error correction).

With error correction,

the number of qubits should be considerably increased

Fortunately, the number required to factor integers using Shor's algorithm is still only

polynomial, and thought to be between L4 and L6, where L is the number of bits in the

number to be factored.

For a 1000-bit number, this implies a need for 1012 to 1018 qubits.

The fault-tolerant computing manual

(after J. Preskill, 1997)

The fault-tolerant Toffoli gate

(proposed by P. Shor and designed by J. Preskill)

The only problem in this scheme is that

the generation of the cat state is not done fault-tolerantly,

and thus one error can cause the entire cat state to be ruined.

The CNOT gates will then propagate this error to many qubits in the block.

The solution to this problem is to verify that the cat state

is indeed a superposition of |00000000... and |11111111.. , before continuing.

D. Aharonov and M. Ben-Or (1999)

A minor problem with the cat state

2. How big can the cat be?

Ψ = ( + )/ Homework

1. Verify your cat + dog state: 2

Consider a classical system of 103 – 105 compass needles

• Uncontrolled rotations due to noise• Manipulations are not exact• Undesired interactions between our needles

By a) applying external fields to individual needles and b) introducing controlled interactions betweens pairs,

we wish to impose a prescribed evolution of the whole system

Quantum versus classical precision

Some trivial remarks:

We fix a coordinate system xyz related to some physical objects, with the z axis pointing towards the Polar Star

* This direction, as well as the angles between our axes cannot be defined exactly

* The orientation of the needle with respect to our axes cannot be defined exactly

*

* Two needles NEVER point in exactly the same direction

* etc, etc

...7071067811,045cos

Apparently, things are not so obvious in the magic world of quantum mechanics!

There is a widespread belief that the |1 and |0 states “in the computational basis” are something absolute, akin to the on/off states of a classical switch, but with the advantage that one can have quantum superpositions of these states

on off

+Ψ = 2-1/2{ }

Quantum versus classical precision

In reality, pure |1 and |0 states can never exist!

Similarly, a classical vector can never point in exactly the z direction.(We simply never know what is the z direction)

Instead of pure |1, we always have |1 + c |0 with some unknown c, where |c| is hopefully small.

The classical statement: the orientation of any vector is known only within a certain precision

is translated into quantum language as:there is always an admixture of unwanted states to any desirable state

Quantum versus classical precision

Thus the (|1 + |0)/21/2 state, and especially the “cat” state

|cat = (|1111111 + |0000000)/21/2

are abstractions, that can never exist in reality!

Quantum versus classical precision

So, when we say “prepare a (|1 + |0)/21/2 state”, we must realize that

what we really can have, is: a|1 + b|0, where |a| and |b| are close to 2-1/2

with a precision not better than our ability to measure angles (usually, much worse)

If we measure the state a|1 + b|0 and get (0), the wavefunction does NOT collapse exactly to |0. Rather it becomes |0 + c|1, where |c| is small

The precision of qubit states is never better than the precision of the direction of a classical vector!

Square root of 2 does not exist in Nature !The measured diagonal of a unit square is equal to 1.4142 ± 0.0007

This remains true even in the Magic World of Quantum Mechanics !

Quotes

(J. Preskill, 1997)

Doubts...

* The existing scheme for fault tolerant quantum computing is an extremely complicated construction. It is not sufficiently clear how it is going to work.

* It is not obvious that the scheme takes properly account of:

- impossibility of preparing exactly |0 and |1 states, or any desired state

- continuous nature of decoherence of all qubits

- continuous nature of gate errors

- N times faster deterioration of the wavefunction of N qubits, compared to the decoherence time for an individual qubit

- the true outcome of imperfect measurements

- the finite duration of gates, measurements, and any action

* Besides, in reality there will always be some correlation in noise for different qubits, systematic errors of gates, undesired interactions between qubits etc. (something beyond the adopted model).

What will be the result of even small effects of this kind?

My own Theorem

Theorem 1: No continuous variable can be equal exactly to zero

Consequence: A continuous variable can not have an exact value ( e.g. √2 )

Example: Any action on the wave function of 1000 qubits with 10300 amplitudes is described by a matrix 10300 × 10300 . You think, that you can apply a matrix, corresponding to a two-qubit gate:

. . . . . . . . . . . . . . . . . . . . . . . . . . . 0 0 0 0 0 0 0 . . . . . . 0 0 1 i -1 0 0 . . . . . . 0 -i 0 1 0 0 0 . . . . . . 0 0 -1 0 i 0 0 . . . . . . 0 i 0 -1 0 0 0 . . . . . . 0 0 0 0 0 0 0 . . . . . . . . . . . . . . . . . . . . . . . . . .

However, according to Theorem 1 this is impossible. You will never have zeros everywhere. In simple words, you can never completely isolate all the other qubits from the action of the applied field

Doubts...

Especially,

undesired interactions between different qubits = death to quantum computing

How small should be such interactions to make large scale quantum computing possible?

- No answer so far …

Maintaining one qubit in its unknown state

Challenge to the QC community:

Provide a detailed sequence of gates and

measurements that allow to maintain the spin

close enough to its initial direction (along x),

once the error per computation step is less

than some ε

I will then check it on my PC

The lessons from building perpetual motion machines of the second kind

This lesson should make us extremely vigilant to the explicit or implicit presence of ideal elements within the error-correcting theoretical schemes.

The basic point of all the projects of perpetual motion machines of the second kind, existing to date and yet to come, is some ideal element, which is not sensitive to thermal noise.

It is sufficient to have just one such element, often deeply hidden within a very complicated construction. Finding and identifying such an ideal element may be a daunting task.

Important differencies: we are doing some work on our system, we do not want to violate the Second Law (not yet).

Similarity: we are attempting to achieve a reversible evolution of a very large many-body system (on a microscopic level) in the presence of noise and using noisy devices.

The idea, that quantum mechanics helps us to do it, looks strange, since all our experience in physics suggests that quantum effects are more fragile than classical ones.

The quantum computer is much more powerful than a classical one. In doing what ?

Answer: in factoring very large numbers (provided that it will work)

What else? - Oh yes: it will do simulations of quantum systems...

“The transistors in our classical computers are becoming smaller and smaller, approaching the atomic scale. The functioning of future devices will be governed by quantum laws. However, quantum behaviour cannot be efficiently simulated by digital computers. Hence, the enormous power of quantum computers will help us to design the future quantum technology”

(A love-song for the military sponsors)

“In addition, efficient quantum simulation will make a revolution in physics, chemistry, and biology”

Plenty time to think about it…

J. Preskill, 1997

A. Steane, 1997

April 2, 2004

Conclusions

It is absolutely incredible, that

by applying external fields, which cannot be calibrated perfectly by doing imperfect measurements and by using converging sequences of “fault-tolerant”, but imperfect, gates

one can continuously protect the grand wavefunction from the random drift of its 10300 amplitudes and moreover make these amplitudes change in a precise and regular manner needed for quantum computations on a time scale greatly exceeding the relaxation time

Summary: we cannot have it and apparently we don’t even need it…

It seems very likely that the (theoretical) success of error-correcting schemes is based on the implicit introduction of ideal elements, like exact |1 and |0 states, and other unrealisticassumptions. In other words, it seems that the math is detached from the physical reality

The claim that one qubit can be kept intact in the presence of noise, should be supported by a detailed description of the needed sequence of manipulations and measurements. A numerical test for a realistic model of imperfections should be done

Factoring large numbers is not a goal that can inspire generations of physicists and engineers. Ideas for other applications, like simulating quantum systems, are vague and far from being obvious