We consider fault-tolerant computation with formulas composed of noisy Boolean gates with two input wires. In our model all gates fail independently of each other and of the input. When a gate fails, it outputs the opposite of the correct output. It is known that if all gates fail with probability at least β2 = (3− √ 7)/4 ≈ 8.856%, faulttolerant computation is not possible. On the other hand, if all gates fail with probability < β2 and is the same for all gates, then fault-tolerant computation is possible. The assumption that all gates fail with exactly the same probability is pretty strong and unrealistic in real-world scenarios. Furthermore, one might be tempted to think that it can be removed easily, since making gates “better” should not hurt. Surprisingly, this is not the case, as we show in this work: there is a constant α2 < β2 such that almost all functions cannot be computed by formulas, if the noise rate of each individual gate is selected adversarially in the range [0, α2]. Hence, while a hardware manufacturer who consistently produces bad gates with noise rate α2 can always achieve reliable computation, a manufacturer who can only ensure that all gates have noise rates at most α2 cannot.
We consider fault-tolerant computation with formulas composed of noisy Boolean gates with two input wires. In our model all gates fail independently of each other and of the input. When a gate fails, it outputs the opposite of the correct output. It is known that if all gates fail with probability at least β2 = (3− √ 7)/4 ≈ 8.856%, faulttolerant computation is not possible. On the other hand, if all gates fail with probability < β2 and is the same for all gates, then fault-tolerant computation is possible. The assumption that all gates fail with exactly the same probability is pretty strong and unrealistic in real-world scenarios. Furthermore, one might be tempted to think that it can be removed easily, since making gates “better” should not hurt. Surprisingly, this is not the case, as we show in this work: there is a constant α2 < β2 such that almost all functions cannot be computed by formulas, if the noise rate of each individual gate is selected adversarially in the range [0, α2]. Hence, while a hardware manufacturer who consistently produces bad gates with noise rate α2 can always achieve reliable computation, a manufacturer who can only ensure that all gates have noise rates at most α2 cannot.
Evans and Pippenger showed in 1998 that e-noisy gates with 2 inputs are universal for arbitrary computation (i.e. can compute any function with bounded error), if all gates fail independently with probability isin and isin0= (3 - radic7)/4 ap8.856%. We show that formulas built from gates with 2 inputs, in which each gate fails with probability at least isin0cannot be universal. Hence, there is a threshold on the tolerable noise for formulas with 2-input gates and it is isin0. We conjecture that the same threshold also holds for circuits.
It is known that epsi-noisy gates with two inputs are universal for arbitrary computation (i.e., can compute any function with bounded error), if all gates fail independently with probability epsi and epsi2= (3 - radic7)/4 ap 8.856%. In this paper, it is shown that this bound is tight for formulas, by proving that gates with two inputs, in which each gate fails with probability at least beta2cannot be universal. Hence, there is a threshold on the tolerable noise for formulas with two-input gates and it is beta2. It is conjectured that the same threshold also holds for circuits.
This thesis is concerned with certain theoretical problems that arise naturally in the context of fault-tolerant quantum computation. Fault-tolerance can be defined as the art of building reliable devices from unreliable components, and is of particular importance for quantum computers that aim to precisely control the dynamics of extremely sensitive quantum systems. A model of quantum computation is a specification of the basic building blocks by which a quantum computation is implemented. The best known model is the quantum circuit model, where computations are implemented by means of unitary quantum gates that are applied to two-level quantum systems known as qubits. In a physical implementation of a quantum circuit, the gates and qubits will inevitably be affected by noise. Fault-tolerant quantum circuits are designed to be resilient against the effects of this noise, provided that it is not too strong. Fault-tolerance in the quantum circuit model is well developed thanks to the theory of quantum error-correcting codes. These codes allow for the correction of small numbers of errors introduced by a variety of noise processes. In a fault-tolerant quantum circuit, qubits are replaced with encoded qubits, and quantum gates with encoded gates that are immediately followed by special quantum circuits for error correction. Provided the rate at which errors occur is below a constant threshold value, the accumulation of errors can be checked so that the correct output of the computation correctly determined. The threshold acts as both a measure of how good the design of a quantum circuit is, and also as a target for experimenters aiming to implement quantum circuits. Much current research in quantum computation is aimed at designing quantum circuits that increase the noise threshold, hopefully to the point where it becomes within the reach of experimenters. A fault-tolerant encoded quantum gate must limit the propagation of errors so that the code’s corrective capabilities are not overwhelmed. It is not so easy to design encoded gates that satisfy this property, and to date only a finite handful of such gates are known. In a faulttolerant quantum circuit, all quantum gates must be decomposed or compiled in terms of those that may be implemented fault-tolerantly. In the first part of this thesis, we present two results that may be applied to this problem of gate compilation. The first is a generic method based on the Solovay-Kitaev theorem that may be applied to all quantum gates, but is most effective for those that act on single qubits. We present the Solovay-Kitaev theorem in its simples known form as an algorithm, together with novel constructions that can be used to implement it. Following this we give a two specialized methods for two-qubit gate decomposition, based on the Cartan decomposition of the Lie group SU(4). The cluster-state model of computation is an alternative to the quantum circuit model, and makes use of quantum measurements and highly entangled cluster-states to implement a quantum computation. The fault-tolerant techniques developed for quantum circuits are not immediately applicable in this model, so in order for it to be a realistic candidate for performing computations we prove that such techniques are possible. In the second part of the thesis we prove that constant fault-tolerance thresholds may be achieved in the cluster-state model, and in particular in an adaption to an optical implementation. Following this we design a complete error correction scheme for optical cluster-state computation, and numerically determine the threshold of this model in the face of the dominant noise models likely to affect such an implementation.
We prove new upper bounds on the tolerable level of noise in a quantum circuit. We consider circuits consisting of unitary k-qubit gates each of whose input wires is subject to depolarizing noise of strength p, as well as arbitrary one-qubit gates that are essentially noise-free. We assume that the output of the circuit is the result of measuring some designated qubit in the final state. Our main result is that for p>1−Θ(1/k), the output of any such circuit of large enough depth is essentially independent of its input, thereby making the circuit useless. For the important special case of k=2, our bound is p>35.7%. Moreover, if the only allowed gate on more than one qubit is the two-qubit CNOT gate, then our bound becomes 29.3%. These bounds on p are numerically better than previous bounds, yet are incomparable because of the somewhat different circuit model that we are using. Our main technique is the use of a Pauli basis decomposition, in which the effects of depolarizing noise are very easy to describe.
We prove new upper bounds on the tolerable level of noise in a quantum circuit. We consider circuits consisting of unitary k-qubit gates each of whose input wires is subject to depolarizing noise of strength p, as well as arbitrary one-qubit gates that are essentially noise-free. We assume that the output of the circuit is the result of measuring some designated qubit in the final state. Our main result is that for p>1-Θ(1/\sqrt{k}), the output of any such circuit of large enough depth is essentially independent of its input, thereby making the circuit useless. For the important special case of k=2, our bound is p>35.7%. Moreover, if the only allowed gate on more than one qubit is the two-qubit CNOT gate, then our bound becomes 29.3%. These bounds on p are notably better than previous bounds, yet are incomparable because of the somewhat different circuit model that we are using. Our main technique is the use of a Pauli basis decomposition, which we believe should lead to further progress in deriving such bounds.
This paper shows that quantum computation can be made fault-tolerant against errors and inaccuracies when η, the probability for an error in a qubit or a gate, is smaller than a constant threshold ηc. This result improves on Shor's result [Proceedings of the 37th Symposium on the Foundations of Computer Science, IEEE, Los Alamitos, CA, 1996, pp. 56–65], which shows how to perform fault-tolerant quantum computation when the error rate η decays polylogarithmically with the size of the computation, an assumption which is physically unreasonable. The cost of making the quantum circuit fault-tolerant in our construction is polylogarithmic in time and space. Our result holds for a very general local noise model, which includes probabilistic errors, decoherence, amplitude damping, depolarization, and systematic inaccuracies in the gates. Moreover, we allow exponentially decaying correlations between the errors both in space and in time. Fault-tolerant computation can be performed with any universal set of gates. The result also holds for quantum particles with p>2 states, namely, p-qudits, and is also generalized to one-dimensional quantum computers with only nearest-neighbor interactions. No measurements, or classical operations, are required during the quantum computation. We estimate the threshold of our construction to be ηc≃10−6, in the best case. By this we show that local noise is in principle not an obstacle for scalable quantum computation. The main ingredient of our proof is the computation on states encoded by a quantum error correcting code (QECC). To this end we introduce a special class of Calderbank–Shor–Steane (CSS) codes, called polynomial codes (the quantum analogue of Reed–Solomon codes). Their nice algebraic structure allows all of the encoded gates to be transversal. We also provide another version of the proof which uses more general CSS codes, but its encoded gates are slightly less elegant. To achieve fault tolerance, we encode the quantum circuit by another circuit by using one of these QECCs. This step is repeated polyloglog many times, each step slightly improving the effective error rate, to achieve the desired reliability. The resulting circuit exhibits a hierarchical structure, and for the analysis of its robustness we borrow terminology from Khalfin and Tsirelson [Found. Phys., 22 (1992), pp. 879–948] and Gács [Advances in Computing Research: A Research Annual: Randomness and Computation, JAI Press, Greenwich, CT, 1989]. The paper is to a large extent self-contained. In particular, we provide simpler proofs for many of the known results we use, such as the fact that it suffices to correct for bit-flips and phase-flips, the correctness of CSS codes, and the fact that two-qubit gates are universal, together with their extensions to higher-dimensional particles. We also provide full proofs of the universality of the sets of gates we use (the proof of universality was missing in Shor's paper). This paper thus provides a self-contained and complete proof of universal fault-tolerant quantum computation in the presence of local noise.