 Award ID(s):
 1718494
 Publication Date:
 NSFPAR ID:
 10064523
 Journal Name:
 IEEE International Symposium on Information Theory
 Sponsoring Org:
 National Science Foundation
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A basic question in the theory of faulttolerant quantum computation is to understand the fundamental resource costs for performing a universal logical set of gates on encoded qubits to arbitrary accuracy. Here we consider qubits encoded with constant space overhead (i.e. finite encoding rate) in the limit of arbitrarily large code distance d through the use of topological codes associated to triangulations of hyperbolic surfaces. We introduce explicit protocols to demonstrate how Dehn twists of the hyperbolic surface can be implemented on the code through constant depth unitary circuits, without increasing the space overhead. The circuit for a given Dehn twist consists of a permutation of physical qubits, followed by a constant depth local unitary circuit, where locality here is defined with respect to a hyperbolic metric that defines the code. Applying our results to the hyperbolic Fibonacci TuraevViro code implies the possibility of applying universal logical gate sets on encoded qubits through constant depth unitary circuits and with constant space overhead. Our circuits are inherently protected from errors as they map local operators to local operators while changing the size of their support by at most a constant factor; in the presence of noisy syndrome measurements, our results suggestmore »

Abstract We study twoqubit circuits over the Clifford+CS gate set, which consists of the Clifford gates together with the controlledphase gate CS = diag(1, 1, 1,
i ). The Clifford+CS gate set is universal for quantum computation and its elements can be implemented faulttolerantly in most errorcorrecting schemes through magic state distillation. Since nonClifford gates are typically more expensive to perform in a faulttolerant manner, it is often desirable to construct circuits that use few CS gates. In the present paper, we introduce an efficient and optimal synthesis algorithm for twoqubit Clifford+CS operators. Our algorithm inputs a Clifford+CS operatorU and outputs a Clifford+CS circuit forU , which uses the least possible number of CS gates. Because the algorithm is deterministic, the circuit it associates to a Clifford+CS operator can be viewed as a normal form for that operator. We give an explicit description of these normal forms and use this description to derive a worstcase lower bound of on the number of CS gates required to$$5{{\rm{log}}}_{2}(\frac{1}{\epsilon })+O(1)$$ $5{\mathrm{log}}_{2}\left(\frac{1}{\u03f5}\right)+O\left(1\right)$ϵ approximate elements of SU(4). Our work leverages a wide variety of mathematical tools that may find further applications in the study of faulttolerant quantum circuits. 
Abstract We study the effectiveness of quantum error correction against coherent noise. Coherent errors (for example, unitary noise) can interfere constructively, so that in some cases the average infidelity of a quantum circuit subjected to coherent errors may increase quadratically with the circuit size; in contrast, when errors are incoherent (for example, depolarizing noise), the average infidelity increases at worst linearly with circuit size. We consider the performance of quantum stabilizer codes against a noise model in which a unitary rotation is applied to each qubit, where the axes and angles of rotation are nearly the same for all qubits. In particular, we show that for the toric code subject to such independent coherent noise, and for minimalweight decoding, the logical channel after error correction becomes increasingly incoherent as the length of the code increases, provided the noise strength decays inversely with the code distance. A similar conclusion holds for weakly correlated coherent noise. Our methods can also be used for analyzing the performance of other codes and faulttolerant protocols against coherent noise. However, our result does not show that the coherence of the logical channel is suppressed in the more physically relevant case where the noise strength is heldmore »

Universal quantum computation requires the implementation of a logical nonClifford gate. In this paper, we characterize all stabilizer codes whose code subspaces are preserved under physical T and T † gates. For example, this could enable magic state distillation with nonCSS codes and, thus, provide better parameters than CSSbased protocols. However, among nondegenerate stabilizer codes that support transversal T, we prove that CSS codes are optimal. We also show that triorthogonal codes are, essentially, the only family of CSS codes that realize logical transversal T via physical transversal T. Using our algebraic approach, we reveal new purelyclassical coding problems that are intimately related to the realization of logical operations via transversal T. Decreasing monomial codes are also used to construct a code that realizes logical CCZ. Finally, we use Ax's theorem to characterize the logical operation realized on a family of quantum ReedMuller codes. This result is generalized to finer angle Zrotations in https://arxiv.org/abs/1910.09333.

Variational Quantum Algorithms (VQAs) rely upon the iterative optimization of a parameterized unitary circuit with respect to an objective function. Since quantum machines are noisy and expensive resources, it is imperative to choose a VQA's ansatz appropriately and its initial parameters to be close to optimal. This work tackles the problem of finding initial ansatz parameters by proposing CAFQA, a Clifford ansatz for quantum accuracy. The CAFQA ansatz is a hardwareefficient circuit built with only Clifford gates. In this ansatz, the initial parameters for the tunable gates are chosen by searching efficiently through the Clifford parameter space via classical simulation, thereby producing a suitable stabilizer state. The stabilizer states produced are shown to always equal or outperform traditional classical initialization (e.g., HartreeFock), and often produce high accuracy estimations prior to quantum exploration. Furthermore, the technique is classically suited since a) Clifford circuits can be exactly simulated classically in polynomial time and b) the discrete Clifford space, while scaling exponentially in the number of qubits, is searched efficiently via Bayesian Optimization. For the Variational Quantum Eigensolver (VQE) task of molecular ground state energy estimation up to 20 qubits, CAFQA's Clifford Ansatz achieves a mean accuracy of near 99%, recovering as muchmore »