The basic principles of fault-tolerant quantum circuit simulations and explicit gadget constructions satisfying properties 1 and 2 (to be discussed in chapter 5) were first given by Shor [51]. In this and in all subsequent work, the properties of fault-tolerant gadgets were discussedsemantically, i.e., in terms of an explicit representation of the errors afflicting the quantum state being processed by these gadgets.
After Gottesman, Preskill and I began thinking about the problem of proving the quantum threshold theorem for distance-3 codes, it quickly became clear that the criterion for fault tolerance would involve extended rectangles whose analysis we found to be difficult semantically. In fact, our definition of goodness had previously been used by Knill, Laflamme and Zurek [52], but their argument was informal and it was not at all clear how to apply it to recursive fault-tolerant simula- tions (to be discussed in chapter 3). During and shortly after the QIP 2005 workshop, Gottesman suggested that we use an insight from [53] to phrase our definition of correctness in terms of ideal decoding circuits; thissyntacticdefinition led to the formulation of lemma 1.
An analysis of fault-tolerant quantum computation for a coherent noise model was first formulated in an insightful paper by Terhal and Burkard [49]. Their analysis considered a noise model where a locality condition is imposed not only on the interaction between the system and the bath, but also on interactions among the degrees of freedom within the bath. The essential ingredient for generalizing the proof in [49] was the formulation of lemma 2 for the Hamiltonian noise model described in §2.2. Lemma 2 for this noise model was proved by Preskill and it was included in [1] with a simplified proof due to Gottesman. The version of this lemma given here applies to general local noise; the proof is essentially the same as in [1].
Chapter 3
Recursive Simulations and the
Quantum Threshold Theorem
3.1
Introduction
In this chapter, I will discuss recursive fault-tolerant quantum circuit simulations and prove the quantum threshold theorem for local noise. The idea of recursive simulations is intuitive: In the previous chapter, I described how any ideal quantum circuit can be simulated using gadgets that perform the computation in an encoded form using some quantum error-correcting code. Further- more, these simulations are fault-tolerant in the sense that the noisy simulation is more accurate than the noisy unencoded computation provided the strength of local noise is below some critical value. But, if for some noise strength the fault-tolerant simulation improves the accuracy of the unencoded computation, we expect that a fault-tolerant simulation of the fault-tolerant simulation will improve it even more! And we can continue applying this idea again and again building a hierarchy of simulations that achieves any desired accuracy. We will then identify the critical noise strength as theaccuracy thresholdfor quantum computation.
By construction, at any level, k, of a recursive fault-tolerant simulation, the simulation will be mapped to the corresponding simulation at levelk−1 if we replace every 1-Rec inside it by a 0-Ga. If we replace every 1-Rec inside a simulation at level k−1 by a 0-Ga, the simulaton at level k−1 will be mapped to the corresponding simulation at levelk−2, etc. Our analysis of the accuracy of recursive fault-tolerant simulations will exploit exactly this property: We will show that the fault- tolerant simulation at levelkwith noise strengthεis equivalent to another fault-tolerant simulation at levelk−1 with atransformednoise strength,ε(1). Then, by applying the same argument a second time, the fault-tolerant simulation at level k−1 with noise strength ε(1) is equivalent to another fault-tolerant simulation at level k−2 with atransformed noise strength, ε(2). After we apply this argumentk times, we will obtain an equivalent simulation at level 0 with noise strength ε(k); but, a level-0 simulation is an execution of the ideal circuit with noisy physical operations! We conclude
that the initial fault-tolerant simulation at levelkis equivalent to an execution of the ideal quantum circuit with operations which have noise strength ε(k). Provided the physical noise strength, ε, is below the accuracy threshold value, ε(k) can be made arbitrarily small, thereby showing that the level-ksimulation can achieve any desired accuracy. This is the outline of our proof of the quantum threshold theorem.
The remaining of this chapter is organized as follows: In §3.2, I discuss the replacement rules that give rise to recursive fault-tolerant simulations. Then, in§3.3, I prove thelevel-reductionlemma which shows that recursive fault-tolerant simulations subject to local noise can be analysed “one level at a time.” Finally, in §3.4, I use this lemma to prove the quantum threshold theorem for local noise. I also discuss a method for obtaining improved rigorous lower bounds on the quantum accuracy threshold by means of a more detailed combinatorial analysis; such an analysis is performed explicitly in chapter 5 for a particular code.