DetrimentalDetrimentalDecoherenceDecoherence

Gil KalaiGil KalaiHebrew University of JerusalemHebrew University of Jerusalem

And Yale UniversityAnd Yale UniversityQEC07, Los Angeles, Dec ’07QEC07, Los Angeles, Dec ’07

HU quantum computing sem. HU quantum computing sem. Jan.‘08Jan.‘08

Prepared forPrepared for QEC07QEC07

First International Conference on Quantum Error Correction

University of Southern CaliforniaUniversity of Southern California , ,

Los Angeles 17-21 December, 2007Los Angeles 17-21 December, 2007..

Revised for HU quantum computer

seminar

Hebrew University of Jerusalem – Thursday’s quantum computation seminar, January 17,

2008

Outline of the talkOutline of the talk

11 . .Quantum computers, noisy quantum Quantum computers, noisy quantum computation and fault tolerance. Examplescomputation and fault tolerance. Examples..

22 . .Detrimental decoherence: conjecturesDetrimental decoherence: conjectures

33 . .Extensions and modelsExtensions and models

44 . .The rate of errorsThe rate of errors..

55 . .Comments on classical noise, Comments on classical noise, computational complexity, possible computational complexity, possible counterexamples, etccounterexamples, etc..

BACKGROUNDBACKGROUND

Quantum ComputersQuantum Computers

Quantum computers (Quantum computers (Deutsch, 85Deutsch, 85) are ) are hypotheticalhypothetical devices based on quantum devices based on quantum physics. Here is a brief description of what physics. Here is a brief description of what they arethey are::

The state of a digital computer having n bits The state of a digital computer having n bits is a string of length n of zeros and ones. As is a string of length n of zeros and ones. As a first step towards quantum computers a first step towards quantum computers we can consider (abstractly) stochastic we can consider (abstractly) stochastic versions of digital computers where the versions of digital computers where the state is a (classical) probability distribution state is a (classical) probability distribution on all such stringson all such strings..

Quantum Computers (cont.)Quantum Computers (cont.) Quantum computers are similar to these Quantum computers are similar to these

(hypothetical) stochastic classical (hypothetical) stochastic classical computers and they work on qubits (say n computers and they work on qubits (say n of them)of them)..

The state of a single qubit q is described by The state of a single qubit q is described by

a unit vectora unit vector

u = a |0> + b |1u = a |0> + b |1 > >

in a two-dimensional complex space U[q]. in a two-dimensional complex space U[q]. We can think of the qubit q as representing We can think of the qubit q as representing '0' with probability |a|'0' with probability |a|22 and '1' with and '1' with probability |b|probability |b|22. .

Quantum Computers (cont.)Quantum Computers (cont.)The state of the entire computer is a The state of the entire computer is a unit vector in the 2unit vector in the 2nn dimensional dimensional tensor product of these vector spaces tensor product of these vector spaces U[q]s for the individual qubitsU[q]s for the individual qubits . .

The state of the computer thus The state of the computer thus represents a probability distribution represents a probability distribution on the 2on the 2nn strings of length n of 0’s strings of length n of 0’s and 1’s.and 1’s.

Quantum Computers (cont.)Quantum Computers (cont.)The evolution of the quantum computer is The evolution of the quantum computer is via ``gates.'' Each gate G operates on k via ``gates.'' Each gate G operates on k qubits, and we can even assume that k qubits, and we can even assume that k equals one or two. Every such gate equals one or two. Every such gate represents a unitary operator on the 2represents a unitary operator on the 2k-k- dimensional tensor product of the spaces dimensional tensor product of the spaces that correspond to these k qubitsthat correspond to these k qubits..

In every cycle of the computer, gates act In every cycle of the computer, gates act in parallel on disjoint sets of qubitsin parallel on disjoint sets of qubits..

Quantum Computers (cont.)Quantum Computers (cont.)Moving from a qubit q at a certain Moving from a qubit q at a certain state to the probability distribution it state to the probability distribution it represents is called a measurementrepresents is called a measurement . .

We can assume that measurements of We can assume that measurements of the qubits that amount to a sampling the qubits that amount to a sampling of 0-1 strings according to the of 0-1 strings according to the distribution these qubits represent, is distribution these qubits represent, is the final step of the computationthe final step of the computation..

Quantum Computation Quantum Computation (BQP)(BQP)

Quantum computers as described earlier, Quantum computers as described earlier, (or according to quite a few alternative (or according to quite a few alternative but computationally equivalent but computationally equivalent descriptions,) are capable of doing descriptions,) are capable of doing everything classical computers do and everything classical computers do and more. The remarkable complexity class more. The remarkable complexity class described by polynomial time quantum described by polynomial time quantum computation is called BQPcomputation is called BQP . .

Quantum Computation Quantum Computation (cont.)(cont.)

Peter Shor (Peter Shor (19941994) proved that factoring an ) proved that factoring an n-digits number has a polynomial time n-digits number has a polynomial time quantum algorithm, hence is in BQPquantum algorithm, hence is in BQP . .

There is evidence that BQP goes well There is evidence that BQP goes well beyond factoring and that NP-complete beyond factoring and that NP-complete

problems are much beyond BQPproblems are much beyond BQP . .

Are quantum computers Are quantum computers feasiblefeasible??

The feasibility of (computationally The feasibility of (computationally superior) quantum computers is one of superior) quantum computers is one of the most exciting (and the most exciting (and clear-cut) ) scientific problems of our timescientific problems of our time..

If feasible, QC may represent an amazing If feasible, QC may represent an amazing new physics reality based on human new physics reality based on human technology. QC being unfeasible may technology. QC being unfeasible may represent quite surprising new insights represent quite surprising new insights in physics theoryin physics theory..

Related issues to QC feasibilityRelated issues to QC feasibility

The feasibility of quantum computers is also The feasibility of quantum computers is also relevant to other issues of considerable relevant to other issues of considerable interest that arose independently (and interest that arose independently (and even earlier). Here is a partial listeven earlier). Here is a partial list::

1 )The evolution of open quantum systems.

2 )The “measurement problem” and other issues in the foundations of quantum mechanics.

Related issues to QC feasibility Related issues to QC feasibility (cont.)(cont.)

3 .The existence of (stable) non abelian anyons.

4 .Thermodynamics, non-equilibrium thermodynamics. (Suggestions for “4th law of thermodynamics”, “superthermal particles”, etc.)

5 .Noise.

The Postulate of NoiseThe Postulate of Noise

An early critique of quantum computers put An early critique of quantum computers put forward in the mid-90s by Landauer, Unruh, forward in the mid-90s by Landauer, Unruh, and others concerned the matter of noiseand others concerned the matter of noise::

The postulate of noise: Quantum The postulate of noise: Quantum systems are noisysystems are noisy..

Understanding the meaning and nature of Understanding the meaning and nature of noise (and the reason for noise) is of great noise (and the reason for noise) is of great importance in this context (as in many importance in this context (as in many others)others) . .

Noisy Quantum Noisy Quantum ComputationComputation

Dealing with the issue of noise Dealing with the issue of noise required three important required three important developments: The first was a formal developments: The first was a formal development of a model of noisy development of a model of noisy quantum computation. This was first quantum computation. This was first carried out by Bernstein and Vazirani carried out by Bernstein and Vazirani ((19931993))..

Noisy Quantum Computation Noisy Quantum Computation (cont.)(cont.)

Noisy quantum computers: in every Noisy quantum computers: in every computer-cycle there are some “storage computer-cycle there are some “storage errors” which describe a certain errors” which describe a certain deterioration of the state of the computer deterioration of the state of the computer compared to its intended state. In compared to its intended state. In addition, the gates are not perfect and this addition, the gates are not perfect and this is expressed by “gate errors”. Of course, is expressed by “gate errors”. Of course, these two types of errors propagate along these two types of errors propagate along

the computationthe computation . .

Quantum Error CorrectionQuantum Error Correction

The second major development (The second major development (Shor, Shor, Steane, 1995Steane, 1995) towards fault-tolerant ) towards fault-tolerant quantum computation was the quantum computation was the discovery of quantum error discovery of quantum error correction codescorrection codes..

The threshold theoremThe threshold theorem

Finally, the threshold theorem (Finally, the threshold theorem (1997; 1997; Aharonov-BenOr, Kitaev, Knill-Aharonov-BenOr, Kitaev, Knill-Laflamme-ZuerkLaflamme-Zuerk) asserts that when ) asserts that when the “noise rate” is small, and the the “noise rate” is small, and the noise is “local”, fault tolerant noise is “local”, fault tolerant quantum computation (FTQC) is quantum computation (FTQC) is possiblepossible . .

Detrimental errors

Detrimental errors are hypothetical forms of errors for noisy quantum computers (and more general open quantum systems) which are damaging for quantum error-correction and quantum fault-tolerance .

Detrimental errors for quantum computers and their effects are described by three conjectures and are discussed in this lecture.

Daniel Gottesman’s picturepicture is worth is worth thousand wordsthousand words

The Classical and Quantum Worlds

Daniel Gottesman

This lecture deals with the “desert of decoherence.”

In this “desert” quantum processes are modelled by “unprotected quantum circuits.”

Examples firstExamples first : :Unprotected

quantum circuits and a simple type

of errors.

Unprotected quantum Unprotected quantum programsprograms

An important example to have in mind is error-An important example to have in mind is error-propagation of unprotected quantum programs or propagation of unprotected quantum programs or circuitscircuits . .

Take the standard model of independent errors and Take the standard model of independent errors and suppose that the error rate is so small that it suppose that the error rate is so small that it accumulates at the end of the computation to a accumulates at the end of the computation to a small constant-rate error. This was first studied by small constant-rate error. This was first studied by UnruhUnruh . .

For such errors we will witness that rather than being For such errors we will witness that rather than being independent the errors will tend to synchronizeindependent the errors will tend to synchronize..

Unprotected quantum Unprotected quantum programs –words of cautionprograms –words of caution

Since the error-propagation of unprotected Since the error-propagation of unprotected quantum circuits serves as a “role model” quantum circuits serves as a “role model” for a damaging noise, it is tempting to for a damaging noise, it is tempting to regard error-propagation as the sort of regard error-propagation as the sort of damaging noise for QECdamaging noise for QEC..

This is not the case! Whatever bad properties This is not the case! Whatever bad properties we would like to consider they should be we would like to consider they should be manifested already for the “new” errors in manifested already for the “new” errors in each computer cycle. When the new errors each computer cycle. When the new errors behave nicely, FTQC deals well with their behave nicely, FTQC deals well with their propagationpropagation . .

Unprotected quantum Unprotected quantum programs –Cavaet 2programs –Cavaet 2

Following Unruh we take the standard model of Following Unruh we take the standard model of independent errors and suppose that the independent errors and suppose that the error rate is so small that it accumulates at error rate is so small that it accumulates at the end of the computation to a small the end of the computation to a small constant-rate errorconstant-rate error..

We conjecture that the incremental (new) We conjecture that the incremental (new) errors themselves behave like the errors themselves behave like the acccumulation of errors in an unprotected acccumulation of errors in an unprotected circuits. This also means that taking small circuits. This also means that taking small rate errors according to the standard noise rate errors according to the standard noise models is only a first approximation to the models is only a first approximation to the behavior of unprotected quantum circuitsbehavior of unprotected quantum circuits..

Our main thesisOur main thesis

Quantum noisy systems are best modeled by unprotected quantum circuits.

A simple class of errorsA simple class of errors

Let WLet Wk k represent the error of changing represent the error of changing the kthe kthth qubit to the fixed state of qubit to the fixed state of maximum entropy. For a 0-1 string x of maximum entropy. For a 0-1 string x of length n let Elength n let Ex x denote the tensor denote the tensor product of error operations: Wproduct of error operations: Wk k when when xxk k = 1 and the identity I= 1 and the identity Ikk

when xwhen xkk = 0. = 0.

For a probability distribution D on all 0-For a probability distribution D on all 0-1 strings of length n let E1 strings of length n let ED D == ΣΣ D(x)E D(x)Exx . .

An even simpler class of errorsAn even simpler class of errors

For most of the lecture we can consider For most of the lecture we can consider just errors of the form Ejust errors of the form EDD . We will . We will mention now an even smaller class. Let w mention now an even smaller class. Let w be a probability distribution on the unit be a probability distribution on the unit interval [0,1]. We can define a probability interval [0,1]. We can define a probability distribution D(w) on 0-1 strings of length distribution D(w) on 0-1 strings of length n in two steps as follows: First we choose n in two steps as follows: First we choose t in [0,1] according to w and then we let t in [0,1] according to w and then we let every xevery xkk =1 with probability t =1 with probability t (independently for different k’s.) (independently for different k’s.)

ConjecturesConjecturesOnOn

DecoherenceDecoherenceFor noisy quantum For noisy quantum

computerscomputers

A: Information leaks for pairs A: Information leaks for pairs of qubitsof qubits

Conjecture [A]: A noisy quantum computer is Conjecture [A]: A noisy quantum computer is subject to error with the property that subject to error with the property that information leaks for two substantially information leaks for two substantially entangled qubits have a substantial positive entangled qubits have a substantial positive correlationcorrelation..

Conjecture [A] refers to part of the overall errors Conjecture [A] refers to part of the overall errors affecting noisy quantum computers. But we affecting noisy quantum computers. But we conjecture that the effect of detrimental errors conjecture that the effect of detrimental errors (described by Conjectures [B] and [C]) cannot (described by Conjectures [B] and [C]) cannot be remedied by errors of a different typebe remedied by errors of a different type . .

B: Error SynchronizationB: Error Synchronization

Error-synchronization refers to a situation Error-synchronization refers to a situation where, while the error rate is small, where, while the error rate is small, there is a substantial probability of there is a substantial probability of errors affecting a large fraction of qubiterrors affecting a large fraction of qubit..

Conjecture [B]: For any noisy quantum Conjecture [B]: For any noisy quantum computer at a highly entangled state computer at a highly entangled state there will be a strong effect of error-there will be a strong effect of error-synchronizationsynchronization..

Approximately-local statesApproximately-local states

A (pure) state of a quantum computer is A (pure) state of a quantum computer is approximately local if it is determined approximately local if it is determined (up to a small error) by the induced (up to a small error) by the induced states of small sets of qubitsstates of small sets of qubits . .

Note that this is a combinatorial and not Note that this is a combinatorial and not a geometric notion. Note also that a geometric notion. Note also that states needed for quantum (many-) states needed for quantum (many-) error corrections are not error corrections are not approximately localapproximately local . .

C: CensorshipC: Censorship

Conjecture [C]: The states of noisy Conjecture [C]: The states of noisy quantum computers are quantum computers are approximately localapproximately local..

D: An extensionD: An extension

A proposed extension of detrimental A proposed extension of detrimental errors to general quantum systems errors to general quantum systems readsreads::

Conjecture [D]: A description (or Conjecture [D]: A description (or prescription) of a noisy quantum system prescription) of a noisy quantum system at a state S is subject to error described at a state S is subject to error described by a quantum operation E that tends to by a quantum operation E that tends to commute with every unitary operator commute with every unitary operator that stabilizes Sthat stabilizes S..

E: The rate of errors

Trying to understand the rate of Trying to understand the rate of detrimental errors leads todetrimental errors leads to : :

Conjecture [E]: Any noisy quantum Conjecture [E]: Any noisy quantum system whose states are described by system whose states are described by a Hilbert space V is subject to noise so a Hilbert space V is subject to noise so that for some K > 0, and for every that for some K > 0, and for every subspace U of V the infinitesimal rate subspace U of V the infinitesimal rate of noise restricted to U is at leastof noise restricted to U is at least

K log (dim U)K log (dim U)..

Detrimental Detrimental DecoherenceDecoherence

For noisy quantum For noisy quantum computerscomputers::

Conjectures [A],[B],[C].

The settingThe setting

As described before, we consider a noisy As described before, we consider a noisy quantum computer whose “intended” state is quantum computer whose “intended” state is pure, and we assume that along the evolution pure, and we assume that along the evolution the overall error, namely the gap between the the overall error, namely the gap between the

ideal state and the actual state is smallideal state and the actual state is small . .

The errors can be described by a unitary The errors can be described by a unitary operator on the computer qubits and the operator on the computer qubits and the “neighborhood qubits” or as a quantum “neighborhood qubits” or as a quantum operation E on the space of density matrices operation E on the space of density matrices for these n qubitsfor these n qubits . .

The setting (cont.)The setting (cont.)

The errors we consider are the “new The errors we consider are the “new errors” in a single computer cycleerrors” in a single computer cycle . .

In the discussion of conjectures [A] and In the discussion of conjectures [A] and [B] we assume for simplicity that the [B] we assume for simplicity that the errors are of the form Eerrors are of the form ED D . .

Conjecture [A]Conjecture [A]

Remember that we restrict ourselves to Remember that we restrict ourselves to errors of the form Eerrors of the form ED D which depend on which depend on a probability distribution on 0-1 a probability distribution on 0-1 strings of length n. The error rate L(a) strings of length n. The error rate L(a) for the kth qubit a is simply the for the kth qubit a is simply the probability that xprobability that xk k =1. If b is the jth =1. If b is the jth qubit, let L(a,b) be the correlation qubit, let L(a,b) be the correlation between the event xbetween the event xkk =1 and the =1 and the event xevent xjj = 1 = 1

Conjecture [A] (cont.)Conjecture [A] (cont.)

For a state T of the quantum computer, a For a state T of the quantum computer, a standard measure of entanglement is the standard measure of entanglement is the mutual informationmutual information

S(a;b) = S(TS(a;b) = S(T|a|a ) + S (T ) + S (T|b |b ) – S(T ) – S(T| {a,b}| {a,b}))((S is the entropy functionS is the entropy function).).

The formal version of conjecture [A] isThe formal version of conjecture [A] is::L(a,b) > K(L(a),L(b)) S(a;b)L(a,b) > K(L(a),L(b)) S(a;b)For general form of errors the formal For general form of errors the formal definition of L(a,b) is more complicated definition of L(a,b) is more complicated but the basic idea is similarbut the basic idea is similar..

A stronger formulation I: A stronger formulation I: Two quditsTwo qudits

Conjecture [A] extends to pairs of Conjecture [A] extends to pairs of qudits rather than pairs of qubits qudits rather than pairs of qubits without changewithout change . .

In this generality it applies to disjoint In this generality it applies to disjoint sets of qubits in a noisy quantum sets of qubits in a noisy quantum computercomputer..

A stronger formulation II: A stronger formulation II: Emergent entanglementEmergent entanglement

Entanglement between qubits can Entanglement between qubits can emerge when we measure other emerge when we measure other qubits and “look at” the results. A qubits and “look at” the results. A strong form of conjecture [A] takes strong form of conjecture [A] takes this into account and replaces this into account and replaces entanglement with a more general entanglement with a more general notion of emergent entanglementnotion of emergent entanglement . .

A stronger formulation III: A stronger formulation III: Many qubitsMany qubits

Another strong form of conjecture [A] Another strong form of conjecture [A] applies to larger sets of qubitsapplies to larger sets of qubits..

B: Error SynchronizationB: Error Synchronization

Suppose that the error rate for every Suppose that the error rate for every qubit is t. For our error models Equbit is t. For our error models ED D this means that the probability that this means that the probability that xxkk =1 is t for every k =1 is t for every k . .

In the standard models of noise the In the standard models of noise the probability that a fraction of (t+a) probability that a fraction of (t+a) qubits are damaged is exponentially qubits are damaged is exponentially small with the number of qubits n for small with the number of qubits n for every a>0every a>0..

B: Error SynchronizationB: Error Synchronization

Error synchronization means that for some t Error synchronization means that for some t which is much larger than s there is a which is much larger than s there is a substantial probability thatsubstantial probability thatxxkk =1 for t or more indices k =1 for t or more indices k..

For example, when w is a probability For example, when w is a probability distribution on [0,1] and we consider the distribution on [0,1] and we consider the distribution Edistribution ED(w)D(w) . The standard models of . The standard models of noise assume that w is a Dirac distribution noise assume that w is a Dirac distribution (supported on one point). We will witness (supported on one point). We will witness error synchronization if the average of w is t error synchronization if the average of w is t but w is supported on much larger real but w is supported on much larger real

numbersnumbers . .

Error SynchronizationError Synchronization??

An aside: Is error synchronization something we can An aside: Is error synchronization something we can really expect in highly correlated systems? Is this really expect in highly correlated systems? Is this something we witness in nature? Two quick something we witness in nature? Two quick remarksremarks::

a) Perhaps we do see error-synchronization even in a) Perhaps we do see error-synchronization even in correlated classical systemscorrelated classical systems . .

b) The hoped-for-argument would be counterfactual. b) The hoped-for-argument would be counterfactual. Highly entangled systems as required in quantum Highly entangled systems as required in quantum computers (will lead to) come along with very computers (will lead to) come along with very strong error synchronization (that we do not often strong error synchronization (that we do not often encounter), which in turn implies that such highly encounter), which in turn implies that such highly entangled states are unrealisticentangled states are unrealistic . .

Conjecture C and Mathematical Conjecture C and Mathematical challengeschallenges

For lack of time we will not attempt to For lack of time we will not attempt to describe formally conjecture C. Once describe formally conjecture C. Once described mathematically a remaining described mathematically a remaining challenge will be to deduce conjectures challenge will be to deduce conjectures [B] and [C] from conjecture [A] and its [B] and [C] from conjecture [A] and its extensions. Errors of the form Eextensions. Errors of the form EDD can can serve as a good starting point. We serve as a good starting point. We would also like to deduce from the would also like to deduce from the conjectures on physical qubits similar conjectures on physical qubits similar statements for protected qubitsstatements for protected qubits!!

Mathematical challenges Mathematical challenges (cont.)(cont.)

It would also be nice to have an It would also be nice to have an entropy based description of error-entropy based description of error-synchronization without referring to synchronization without referring to the expansion in terms of tensor-the expansion in terms of tensor-product of Pauli operatorsproduct of Pauli operators . .

An extension toAn extension to

general quantum general quantum systemssystems

If the conjectures we propose are If the conjectures we propose are correct they should represent a correct they should represent a property of noise which is not limited property of noise which is not limited to quantum computersto quantum computers..

However our conjectures [A], [B] and However our conjectures [A], [B] and [C] strongly rely on the tensor [C] strongly rely on the tensor product structure of the Hilbert space product structure of the Hilbert space describing the states of quantum describing the states of quantum computerscomputers . .

Conjecture [D]Conjecture [D]

Conjecture [D]: A description (or Conjecture [D]: A description (or prescription) of a noisy quantum prescription) of a noisy quantum system at a state S is subject to error system at a state S is subject to error described by a quantum operation E described by a quantum operation E that tends to commute with every that tends to commute with every unitary operator that stabilizes Sunitary operator that stabilizes S..

Conjecture [D]: why and what

The rationale behind [D] goes as follows :

Our conjectures suggest that if E represents the error for state S and E' represents the error for state U(S), for a unitary operator U on V, then E' will be ``close'' to U-1EU. In particular, this implies that if U(S)=S then E' is ``close'' to U-1EU; hence UE is ``close'' to EU.

Conjecture [D]: why and what (cont.)

Greg Kuperberg pointed out that at a thermodynamics equilibrium a certain limiting error E will actually commute with every U that stabilizes S. One possible way to regard Conjecture [D] is as a statement referring to non-equilibrium thermodynamics.

ModelsModels

ModelsModels

Models exhibiting conjectures [A] and [B] Models exhibiting conjectures [A] and [B] should exhibit them already for the should exhibit them already for the storage-errors (or gate-errors). The new storage-errors (or gate-errors). The new errors may be represented by a rapid errors may be represented by a rapid quantum circuitquantum circuit . .

Such models may be created by pushing Such models may be created by pushing the model of Aharonov, Kitaev and Preskill the model of Aharonov, Kitaev and Preskill a little further. Error synchronization a little further. Error synchronization arises in a paper by Klesse and Frankarises in a paper by Klesse and Frank..Here is a toy model that can be examinedHere is a toy model that can be examined..

A toy modelA toy model

There are no gate errors. Consider the There are no gate errors. Consider the graph G whose vertices are the qubits graph G whose vertices are the qubits and whose edges are qubits that and whose edges are qubits that occur in a gate. Edges are labeled by occur in a gate. Edges are labeled by the gate imperfectionthe gate imperfection . .

The storage error is described by EThe storage error is described by ED D

where the probability distribution D is where the probability distribution D is given by an Ising model on the graph given by an Ising model on the graph G based on these gate-imperfectionsG based on these gate-imperfections . .

Consequences of Consequences of Detrimental Detrimental DecoherenceDecoherence::

Computational Computational complexitycomplexity

How damaging are low rate How damaging are low rate detrimental errorsdetrimental errors

I would expect that detrimental errors will fail I would expect that detrimental errors will fail current methods for fault tolerance and current methods for fault tolerance and quantum linear error correctionquantum linear error correction..

On the other hand, low rate detrimental errors On the other hand, low rate detrimental errors may still allow (with polynomial or quasi-may still allow (with polynomial or quasi-polynomial overhead) classical polynomial overhead) classical computations and log-depth quantum computations and log-depth quantum computationcomputation..

Log-depth quantum computation (+ classical Log-depth quantum computation (+ classical computation) is good enough for polynomial-computation) is good enough for polynomial-time factoringtime factoring . .

Aaronson’s Shor/sure challengeAaronson’s Shor/sure challenge

Scott Aaronson suggested a very nice Scott Aaronson suggested a very nice challenge: Propose a restriction on QC that challenge: Propose a restriction on QC that will not allow polynomial time factoring will not allow polynomial time factoring and would not violate empirical resultsand would not violate empirical results..

This looks very difficult. I am not aware of This looks very difficult. I am not aware of methods that will allow a reduction to a methods that will allow a reduction to a computational power below log-depth computational power below log-depth quantum computing, when the error-rate quantum computing, when the error-rate is smallis small . .

The rate of errorsThe rate of errors

And decoherence And decoherence free subspacesfree subspaces

High-rate errorsHigh-rate errors

A major obstacle for fault tolerance is A major obstacle for fault tolerance is high error-ratehigh error-rate . .

When we consider the standard models When we consider the standard models and perceptions regarding noise there and perceptions regarding noise there is not much reason to believe that the is not much reason to believe that the error rate (for individual qubits) will error rate (for individual qubits) will increase in terms of the number of increase in terms of the number of qubits of the computerqubits of the computer..

If we examine unprotected quantum If we examine unprotected quantum circuits things are differentcircuits things are different..

The rate of errors for The rate of errors for unprotected quantum circuitsunprotected quantum circuits

For unprotected quantum circuits, not For unprotected quantum circuits, not only do the errors tend to synchronize, only do the errors tend to synchronize, but the error-propagation causes the but the error-propagation causes the error-rate itself to depend on the error-rate itself to depend on the complexity of the target state. This complexity of the target state. This

may suggest a tentative conjecturemay suggest a tentative conjecture : :

Conjecture [E] (v.1) The rate of Conjecture [E] (v.1) The rate of detrimental errors in a noisy quantum detrimental errors in a noisy quantum computer is higher for highly entangled computer is higher for highly entangled statesstates..

Critique of the tentative Critique of the tentative conjectureconjecture

Conjecture [E] (v. 1) is quite problematic. If Conjecture [E] (v. 1) is quite problematic. If QEC fails we can indeed expect (as the QEC fails we can indeed expect (as the effect of error–propagation) that the error effect of error–propagation) that the error rate will increase when we prepare rate will increase when we prepare complicated statescomplicated states . .

However, as is, this conjecture adds little more However, as is, this conjecture adds little more to the conjecture “QEC fails”. Moreover, to the conjecture “QEC fails”. Moreover, unlike conjectures [A] and [B], where both unlike conjectures [A] and [B], where both the assumptions and conclusions depended the assumptions and conclusions depended on the tensor product structure, here the on the tensor product structure, here the conclusion does not depend on this conclusion does not depend on this structure. Let’s try another avenuestructure. Let’s try another avenue..

Rate of errors – take 2Rate of errors – take 2

The common convention about the rate of noiseThe common convention about the rate of noise is that in every computer cycle there is a is that in every computer cycle there is a positive small probability for every qubit to be positive small probability for every qubit to be damaged. The infinitesimal rate of errors for k damaged. The infinitesimal rate of errors for k qubits taken together is just k times that of a qubits taken together is just k times that of a single qubit error-ratesingle qubit error-rate . .

Conjecture [E] (v.2) : Any noisy quantum system Conjecture [E] (v.2) : Any noisy quantum system whose states are described by a Hilbert space whose states are described by a Hilbert space V is subject to noise so that for some K>0, for V is subject to noise so that for some K>0, for every subspace U of V, the infinitesimal rate every subspace U of V, the infinitesimal rate of noise restricted to U is at leastof noise restricted to U is at least

K log (dim U)K log (dim U)..

Rate of errors – take 2 Rate of errors – take 2 (cont.)(cont.)

This (very strong and rather general) This (very strong and rather general) conjecture [E] can be regarded as a conjecture [E] can be regarded as a formulation of the formulation of the postulate of noisepostulate of noise that runs directly against the idea of that runs directly against the idea of decoherence-free subspaces. It decoherence-free subspaces. It agrees with the behavior we observe agrees with the behavior we observe for unprotected quantum circuitsfor unprotected quantum circuits . .

Conjecture [E] may damage even log-Conjecture [E] may damage even log-depth quantum computationdepth quantum computation..

Conjecture [E] (cont.) Conjecture [E] (repeated): Any noisy quantum Conjecture [E] (repeated): Any noisy quantum system whose states are described by a Hilbert system whose states are described by a Hilbert space V is subject to noise so that for some space V is subject to noise so that for some K>0, for every subspace U of V the K>0, for every subspace U of V the infinitesimal rate of noise restricted to U is at infinitesimal rate of noise restricted to U is at leastleast

K log (dim U)K log (dim U)..

In order to exclude decoherence free subspaces, In order to exclude decoherence free subspaces, Conjecture [E] would imply error-Conjecture [E] would imply error-synchronization. Moreover, the rate (for a synchronization. Moreover, the rate (for a single qubit) of highly synchronized errors will single qubit) of highly synchronized errors will scale up linearlyscale up linearly with the number of qubits with the number of qubits..

The rate of errors (cont.)

We can also expect that the rate K of We can also expect that the rate K of detrimental errors for a prescribed (or detrimental errors for a prescribed (or described) evolution of a quantum described) evolution of a quantum system, depends on a measure of system, depends on a measure of non-commutativity between the space non-commutativity between the space P of unitary operators leading to the P of unitary operators leading to the state from the initial state, and the state from the initial state, and the space F of unitary operators leading space F of unitary operators leading from the state to the terminal statefrom the state to the terminal state..

Difficulties and Difficulties and potential counter potential counter

examplesexamplesA few difficulties and potential counterexamples for A few difficulties and potential counterexamples for

conjectures [A], [B] and [C] are describedconjectures [A], [B] and [C] are described . .

Two photonsTwo photons

Errors for two far-away entangled Errors for two far-away entangled photons are not correlatedphotons are not correlated..

((So the rate of detrimental errors in So the rate of detrimental errors in this case is 0this case is 0 ). ).

Classical fault toleranceClassical fault tolerance

If fault tolerant quantum computing If fault tolerant quantum computing fails, how is it that fault tolerance fails, how is it that fault tolerance classical computing prevailsclassical computing prevails??

The formal versions (and wordings) of the conjectures are “tailored” to avoid these two difficulties .

Still these are genuine difficulties that should be kept in mind.

SuperconductivitySuperconductivity

Is superconductivity a counter Is superconductivity a counter exampleexample ? ?

((Or, at least, isn’t it true that similar Or, at least, isn’t it true that similar pessimistic conjectures could have been pessimistic conjectures could have been raised regarding superconductivity had it raised regarding superconductivity had it

not been witnessednot been witnessed )? )?

2n bosons

(This is a potential counter example I cooked by myself ).A state of 2n bosons

each having a ground state |0> and an excited state |1> so that each state has occupation number precisely n appears to violate Conjecture C. Is it realistic? )If the occupation number has a normal distribution this is OK.(

nonabelyons

Stable non abelian anyons, which some expect to witness rather soon, run against our conjectures.

(There is much theoretical and empirical effort regarding creation, detection and applications of non abelyan anyons. I am not aware of a systematic theoretic study for why they cannot be created).

Fermi fermionsBose bosonsAny anyons

ConclusionConclusion : :The story we try to tell

ConclusionWe are trying to describe a story of our physical world without quantum error-correction, decoherence-free subspaces and perhaps even without quantum computing which goes beyond classical computing. (But, of course, a story well within quantum mechanics.)

We start telling it in a very special way - just about two qubits ([A]) so that it could be tested easily for small devices. But we also tried to tell it in a very general way ([D] and [E]) which goes beyond quantum computers .

Conclusion (cont.)

We try to tell the story as formally and as explicitly as possible (this makes for most of the effort and there is a way to go), and to make it quantitative. We tried to make our story bold as to make it easy to refute. ([C] and [E] are the boldest. Does [E] violate the empirical results presented by Laflamme?) We point out surprising aspects (Error-synchronization [B]) and we consider some analogies (classical noise). We attempt to

make it into an elegant story .

Of course, at the end it also has to be correct...

Clarke’s three laws of prediction

1 )When a distinguished but elderly scientist states that something is possible, he is almost certainly right. When he states that something is impossible, he is very probably wrong .

2 )The only way of discovering the limits of the possible is to venture a little way past them into the impossible.

3 )Any sufficiently advanced technology is indistinguishable from magic.

Anyway, it is fun. Thank you!