Detrimental Decoherence

1. DetrimentalDecoherence Gil Kalai Hebrew University of Jerusalem And Yale University QEC07, Los Angeles, Dec ’07 HU quantum computing sem. Jan.‘08

2. Prepared forQEC07 • First International Conference on Quantum Error Correction University of Southern California, Los Angeles 17-21 December, 2007.

3. 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.

4. BACKGROUND

5. Quantum Computers Quantum computers are hypotheticaldevices 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.

6. 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.

7. Quantum Computers (cont.) The state of the entire computer is a unit vector in the 2n 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 2n strings of length n of 0’s and 1’s.

8. 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 2k- dimensional 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.

9. 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.

10. 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.

11. 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.

12. 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.

13. 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.

14. Related issues to QC feasibility (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.

15. The Postulate of Noise An early critique of quantum computers put forward in the mid-90s 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).

16. 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.

17. Noisy Quantum Computation (cont.) Noisy quantum computers: in every computer-cycle 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.

18. 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.

19. 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.

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

21. 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.

22. 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 error-propagation 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.

23. 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 Ikwhen xk = 0. For a probability distribution D on all 0-1 strings of length n let ED =Σ D(x)Ex .

24. 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.)

25. Conjectures On Decoherence For noisy quantum computers

26. 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.

27. 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 error-synchronization.

28. 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.

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

30. 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.

31. 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).

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

33. 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.

34. 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 .

35. 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

36. 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.

37. 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.

38. 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.

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

40. 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.

41. 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.

42. 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.

43. 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!

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

45. An extension to general quantum systems

46. 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.

47. Conjecture [D] 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.

48. 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.

49. 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.

50. Models