Detrimental Decoherence Gil Kalai Hebrew University of Jerusalem

Detrimental Decoherence Gil Kalai Hebrew University of Jerusalem And Yale University QEC 07, Los Angeles, Dec ’ 07 HU quantum computing sem. Jan. ‘ 08

Prepared for QEC 07 First International Conference on Quantum Error Correction University of Southern California, Los Angeles 17 -21 December, 2007. u

Outline of the talk 1. Quantum computers, noisy quantum computation and fault tolerance. Examples. 2. Detrimental decoherence: conjectures 3. Extensions and models 4. The rate of errors. 5. Comments on classical noise, computational complexity, possible counterexamples, etc.

BACKGROUND

Quantum Computers Quantum computers are hypothetical devices based on quantum physics. Here is a brief description of what they are: The state of a digital computer having n bits is a string of length n of zeros and ones. As a first step towards quantum computers we can consider (abstractly) stochastic versions of digital computers where the state is a (classical) probability distribution on all such strings.

Quantum Computers (cont. ) Quantum computers are similar to these (hypothetical) stochastic classical computers and they work on qubits (say n of them). The state of a single qubit q is described by a unit vector u = a |0> + b |1> in a two-dimensional complex space U[q]. We can think of the qubit q as representing '0' with probability |a|2 and '1' with probability |b|2.

Quantum Computers (cont. ) The state of the entire computer is a unit vector in the 2 n dimensional tensor product of these vector spaces U[q]s for the individual qubits. The state of the computer thus represents a probability distribution on the 2 n strings of length n of 0’s and 1’s.

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

Quantum Computers (cont. ) Moving from a qubit q at a certain state to the probability distribution it represents is called a measurement. We can assume that measurements of the qubits that amount to a sampling of 0 -1 strings according to the distribution these qubits represent, is the final step of the computation.

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

Quantum Computation (cont. ) Peter Shor proved that factoring an n-digits number has a polynomial time quantum algorithm, hence is in BQP. There is evidence that BQP goes well beyond factoring and that NP-complete problems are much beyond BQP.

Are quantum computers feasible? The feasibility of (computationally superior) quantum computers is one of the most exciting (and clear-cut) scientific problems of our time. If feasible, QC may represent an amazing new physics reality based on human technology. QC being unfeasible may represent quite surprising new insights in physics theory.

Related issues to QC feasibility The feasibility of quantum computers is also relevant to other issues of considerable interest that arose independently (and even 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 (cont. ) 3. The existence of (stable) non abelian anyons. 4. Thermodynamics, non-equilibrium thermodynamics. (Suggestions for “ 4 th law of thermodynamics”, “superthermal particles”, etc. ) 5. Noise.

The Postulate of Noise An early critique of quantum computers put forward in the mid-90 s by Landauer, Unruh, and others concerned the matter of noise: The postulate of noise: Quantum systems are noisy. Understanding the meaning and nature of noise (and the reason for noise) is of great importance in this context (as in many others).

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

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

QEC and the threshold theorem The second major development towards fault-tolerant quantum computation was the discovery of quantum error correction codes. Finally, the threshold theorem asserts that when the “noise rate” is small, and the noise is “local”, fault tolerant quantum computation (FTQC) is possible.

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.

Examples first: Unprotected quantum circuits and a simple type of errors.

Unprotected quantum programs An important example to have in mind is error-propagation of unprotected quantum programs or circuits. Take the standard model of independent errors and suppose that the error rate is so small that it accumulates at the end of the computation to a small constant-rate error. This was first studied by Unruh. For such errors we will witness that rather than being independent the errors will tend to synchronize.

Unprotected quantum programs –words of caution Since the error-propagation of unprotected quantum circuits serves as a “role model” for a damaging noise, it is tempting to regard errorpropagation as the sort of damaging noise for QEC. This is not the case! Whatever bad properties we would like to consider they should be manifested already for the “new” errors in each computer cycle. When the new errors behave nicely, FTQC deals well with their propagation.

A simple class of errors Let Wk represent the error of changing the kth qubit to the fixed state of maximum entropy. For a 0 -1 string x of length n let Ex denote the tensor product of error operations: Wk when xk = 1 and the identity Ik when xk = 0. For a probability distribution D on all 0 -1 strings of length n let ED = Σ D(x)Ex.

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

Conjectures On Decoherence For noisy quantum computers

A: Information leaks for pairs of qubits Conjecture [A]: A noisy quantum computer is subject to error with the property that information leaks for two substantially entangled qubits have a substantial positive correlation. Conjecture [A] refers to part of the overall errors affecting noisy quantum computers. But we conjecture that the effect of detrimental errors (described by Conjectures [B] and [C]) cannot be remedied by errors of a different type.

B: Error Synchronization Error-synchronization refers to a situation where, while the error rate is small, there is a substantial probability of errors affecting a large fraction of qubit. Conjecture [B]: For any noisy quantum computer at a highly entangled state there will be a strong effect of errorsynchronization.

Approximately-local states A (pure) state of a quantum computer is approximately local if it is determined (up to a small error) by the induced states of small sets of qubits. Note that this is a combinatorial and not a geometric notion. Note also that states needed for quantum (many-) error corrections are not approximately local.

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

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

E: The rate of errors Trying to understand the rate of detrimental errors leads to: Conjecture [E]: Any noisy quantum system whose states are described by a Hilbert space V is subject to noise so that for some K > 0, and for every subspace U of V the infinitesimal rate of noise restricted to U is at least K log (dim U).

Detrimental Decoherence For noisy quantum computers: Conjectures [A], [B], [C].

The setting As described before, we consider a noisy quantum computer whose “intended” state is pure, and we assume that along the evolution the overall error, namely the gap between the ideal state and the actual state is small. The errors can be described by a unitary operator on the computer qubits and the “neighborhood qubits” or as a quantum operation E on the space of density matrices for these n qubits.

The setting (cont. ) The errors we consider are the “new errors” in a single computer cycle. In the discussion of conjectures [A] and [B] we assume for simplicity that the errors are of the form ED.

Conjecture [A] Remember that we restrict ourselves to errors of the form ED which depend on a probability distribution on 0 -1 strings of length n. The error rate L(a) for the kth qubit a is simply the probability that xk =1. If b is the jth qubit, let L(a, b) be the correlation between the event xk =1 and the event xj = 1

Conjecture [A] (cont. ) For a state T of the quantum computer, a standard measure of entanglement is the mutual information S(a; b) = S(T|a ) + S (T|b ) – S(T| {a, b}) (S is the entropy function. ) The formal version of conjecture [A] is: L(a, b) > K(L(a), L(b)) S(a; b) For general form of errors the formal definition of L(a, b) is more complicated but the basic idea is similar.

A stronger formulation I: Two qudits Conjecture [A] extends to pairs of qudits rather than pairs of qubits without change. In this generality it applies to disjoint sets of qubits in a noisy quantum computer.

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

A stronger formulation III: Many qubits Another strong form of conjecture [A] applies to larger sets of qubits.

B: Error Synchronization Suppose that the error rate for every qubit is t. For our error models ED this means that the probability that xk =1 is t for every k. In the standard models of noise the probability that a fraction of (t+a) qubits are damaged is exponentially small with the number of qubits n for every a>0.

B: Error Synchronization Error synchronization means that for some t which is much larger than s there is a substantial probability that xk =1 for t or more indices k. For example, when w is a probability distribution on [0, 1] and we consider the distribution ED(w). The standard models of noise assume that w is a Dirac distribution (supported on one point). We will witness error synchronization if the average of w is t but w is supported on much larger real numbers.

Error Synchronization? An aside: Is error synchronization something we can really expect in highly correlated systems? Is this something we witness in nature? Two quick remarks: a) Perhaps we do see error-synchronization even in correlated classical systems. b) The hoped-for-argument would be counterfactual. Highly entangled systems as required in quantum computers (will lead to) come along with very strong error synchronization (that we do not often encounter), which in turn implies that such highly entangled states are unrealistic.

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

Mathematical challenges (cont. ) It would also be nice to have an entropy based description of errorsynchronization without referring to the expansion in terms of tensor-product of Pauli operators.

An extension to general quantum systems

If the conjectures we propose are correct they should represent a property of noise which is not limited to quantum computers. However our conjectures [A], [B] and [C] strongly rely on the tensor product structure of the Hilbert space describing the states of quantum computers.

Conjecture [D]: A description (or prescription) of a noisy quantum system at a state S is subject to error described by a quantum operation E that tends to commute with every unitary 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-1 EU. In particular, this implies that if U(S)=S then E' is ``close'' to U-1 EU; 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 nonequilibrium thermodynamics.

Models

Models exhibiting conjectures [A] and [B] should exhibit them already for the storage-errors (or gate-errors). Such models may be created by pushing the model of Aharonov, Kitaev and Preskill a little further. Error synchronization arises in a paper by Klesse and Frank. Here is a toy model that can be examined.

A toy model There are no gate errors. Consider the graph G whose vertices are the qubits and whose edges are qubits that occur in a gate. Edges are labeled by the gate imperfection. The storage error is described by ED where the probability distribution D is given by an Ising model on the graph G based on these gate-imperfections.

Consequences of Detrimental Decoherence: Computational complexity

How damaging are low rate detrimental errors I would expect that detrimental errors will fail current methods for fault tolerance and quantum linear error correction. On the other hand, low rate detrimental errors may still allow (with polynomial or quasipolynomial overhead) classical computations and log-depth quantum computation. Log-depth quantum computation (+ classical computation) is good enough for polynomialtime factoring.

Aaronson’s Shor/sure challenge Scott Aaronson suggested a very nice challenge: Propose a restriction on QC that will not allow polynomial time factoring and would not violate empirical results. This looks very difficult. I am not aware of methods that will allow a reduction to a computational power below log-depth quantum computing, when the error-rate is small.

The rate of errors (preliminary and tentative) Especially for QEC 07

High-rate errors A major obstacle for fault tolerance is high error-rate. When we consider the standard models and perceptions regarding noise there is not much reason to believe that the error rate (for individual qubits) will increase in terms of the number of qubits of the computer. If we examine unprotected quantum circuits things are different.

The rate of errors for unprotected quantum circuits For unprotected quantum circuits, not only do the errors tend to synchronize, but the error-propagation causes the error-rate itself to depend on the complexity of the target state. This may suggest a tentative conjecture: Conjecture [E] (v. 1) The rate of detrimental errors in a noisy quantum computer is higher for highly entangled states.

Critique of the tentative conjecture Conjecture [E] (v. 1) is quite problematic. If QEC fails we can indeed expect (as the effect of error–propagation) that the error rate will increase when we prepare complicated states. However, as is, this conjecture adds little more to the conjecture “QEC fails”. Moreover, unlike conjectures [A] and [B], where both the assumptions and conclusions depended on the tensor product structure, here the conclusion does not depend on this structure. Let’s try another avenue.

Rate of errors – take 2 The common convention about the rate of noise is that in every computer cycle there is a positive small probability for every qubit to be damaged. The infinitesimal rate of errors for k qubits taken together is just k times that of a single qubit error-rate. Conjecture [E] (v. 2) : Any noisy quantum system whose states are described by a Hilbert space V is subject to noise so that for some K>0, for every subspace U of V, the infinitesimal rate of noise restricted to U is at least K log (dim U).

Rate of errors – take 2 (cont. ) This (very strong and rather general) conjecture [E] can be regarded as a formulation of the postulate of noise that runs directly against the idea of decoherence-free subspaces. It agrees with the behavior we observe for unprotected quantum circuits. Conjecture [E] may damage even logdepth quantum computation.

Conjecture [E] (cont. ) Conjecture [E] (repeated): Any noisy quantum system whose states are described by a Hilbert space V is subject to noise so that for some K>0, for every subspace U of V the infinitesimal rate of noise restricted to U is at least K log (dim U). In order to exclude decoherence free subspaces, Conjecture [E] would imply error-synchronization. Moreover, the rate (for a single qubit) of highly synchronized errors will scale up linearly with the number of qubits.

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

Difficulties and potential counter examples A few difficulties and potential counterexamples for conjectures [A], [B] and [C] are described.

Two photons Errors for two far-away entangled photons are not correlated. (So the rate of detrimental errors in this case is 0. )

Classical fault tolerance If fault tolerant quantum computing fails, how is it that fault tolerance classical 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.

Superconductivity Is superconductivity a counter example? (Or, at least, isn’t it true that similar pessimistic conjectures could have been raised regarding superconductivity had it not been witnessed? )

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

Conclusion: The story we try to tell

Conclusion We 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 here 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!