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arXiv:1309.3943 · Physical Review A

Pseudorandom circuits from Clifford-plus- T gates

Yaakov S. Weinstein

Published Dec 2, 2013• 4 citations
arXivHTMLPDF

Abstract

We explore the implementation of pseudorandom single-qubit rotations and multiqubit pseudorandom circuits constructed only from Clifford gates and the $T$ gate, a phase rotation of $\ensuremath{\pi}/4$. Such a gate set would be appropriate for computations performed in a fault tolerant setting. For single-qubit rotations the distribution of parameters found for unitaries constructed from Clifford plus $T$ quickly approaches that of random rotations and requires significantly fewer gates than the construction of arbitrary single-qubit rotations. For Clifford-plus-$T$ pseudorandom circuits we find an exponential convergence to a random matrix element distribution and a Gaussian convergence to the higher-order moments of the matrix element distribution. In addition, the nearest-neighbor eigenangle statistics distribution almost immediately converges to that of random unitary matrices. All of these convergence rates are found to be insensitive to the number of qubits.

DOI
10.1103/physreva.88.062303

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We explore the implementation of pseudorandom single-qubit rotations and multiqubit pseudorandom circuits constructed only from Clifford gates and the TTT gate, a phase rotation of \ensuremathπ/4\ensuremath{\pi}/4\ensuremathπ/4. Such a gate set would be appropriate for computations performed in a fault tolerant setting. For single-qubit rotations the distribution of parameters found for unitaries constructed from Clifford plus TTT quickly approaches that of random rotations and requires significantly fewer gates than the construction of arbitrary single-qubit rotations. For Clifford-plus-TTT pseudorandom circuits we find an exponential convergence to a random matrix element distribution and a Gaussian convergence to the higher-order moments of the matrix element distribution. In addition, the nearest-neighbor eigenangle statistics distribution almost immediately converges to that of random unitary matrices. All of these convergence rates are found to be insensitive to the number of qubits.
Building generic gates from a restricted gate set is a difficult but important problem, especially in the noisy regime where only a limited set of noise-resistant gates are available, e.g., fault-tolerant Clifford gates (generated by Hadamard, phase, and gates) and fault-tolerant Toffoli gates. The Toffoli is also often used as a building block of many algorithms and will need to be constructed if not directly available. This makes Clifford + Toffoli an attractive gate set for building generic gates. In this Letter we give a simple and efficient algorithm for building an approximate single-qubit rotation using only Clifford + Toffoli , which in turn enables any generic single-qubit unitary. An important difference compared to earlier attempts is that the use of the Toffoli allows us to use simple rounding as opposed to a complicated approximation algorithm. The resulting gate array does not rely on repeatedly applying a fixed rotation, but immediately applies a rotation R θ * that is ε -close to the desired rotation R θ , with a success probability strictly greater than 1 / 2 . It can be rerun upon failure, giving an expected number of repetitions strictly less than 2, an expected Toffoli count strictly less than 4 ⌈ log 1 ε ⌉ , an expected gate depth strictly less than 4 ⌈ log 1 ε ⌉ + 6 , and uses 2 ⌈ log 1 ε ⌉ ancillas. The small circuit depth of our construction enables low-noise gates on existing quantum computational devices, and allows for arbitrary precision using a very modest number of ancillas. Published by the American Physical Society 2024
Pseudorandom circuits generate quantum states and unitary operators which are approximately distributed according to the unitarily invariant Haar measure. We explore how several design parameters affect the efficiency of pseudorandom circuits, with the goal of identifying relevant tradeoffs and optimizing convergence. The parameters we explore include the choice of single- and two-qubit gates, the topology of the underlying physical qubit architecture, the probabilistic application of two-qubit gates, as well as circuit size, initialization, and the effect of control constraints. Building on the equivalence between pseudorandom circuits and approximate ttt-designs, a Markov matrix approach is employed to analyze asymptotic convergence properties of pseudorandom second-order moments to a 2-design. Quantitative results on the convergence rate as a function of the circuit size are presented for qubit topologies with a sufficient degree of symmetry. Our results may be useful towards optimizing the efficiency of random state and operator generation.
Abstract We study two-qubit circuits over the Clifford+CS gate set, which consists of the Clifford gates together with the controlled-phase gate CS = diag(1, 1, 1, i). The Clifford+CS gate set is universal for quantum computation and its elements can be implemented fault-tolerantly in most error-correcting schemes through magic state distillation. Since non-Clifford gates are typically more expensive to perform in a fault-tolerant 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 two-qubit Clifford+CS operators. Our algorithm inputs a Clifford+CS operator U and outputs a Clifford+CS circuit for U, 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 worst-case lower bound of 5log2(1ϵ)+O(1)5{{\rm{log}}}_{2}(\frac{1}{\epsilon })+O(1)5log2​(ϵ1​)+O(1) 5 log 2 ( 1 ϵ ) + O ( 1 ) on the number of CS gates required to ϵ-approximate elements of SU(4). Our work leverages a wide variety of mathematical tools that may find further applications in the study of fault-tolerant quantum circuits.
The Toffoli gate is an important universal quantum gate, and will alongside the Clifford gates be available in future fault-tolerant quantum computing hardware. Many quantum algorithms rely on performing arbitrarily small single-qubit rotations for their function, and these rotations may also be used to construct any unitary from a limited (but universal) gate set. How to carry out such rotations is then of significant interest. In this work, we evaluate the performance of a recently proposed single-qubit rotation algorithm using the Clifford plus Toffoli gate set by implementation of a one-shot version on both a real and a simulated quantum computer. We test the algorithm under various simulated noise levels using a per-qubit depolarizing error noise model and examine how the probabilities and process fidelities are affected. We then conduct live runs and find that the results reasonably match the simulated results. We also attempt to model the hardware noise by combining a number of noise models, matching the results to results of the live runs to approximate the hardware noise. Our results suggest that the algorithm will perform well for up to 1 % noise under the noise models we chose. We further posit the use of our algorithm as a benchmark for quantum processing units, given that it has a low complexity that is easy to fine-tune in small steps. We provide details for how to do this.
We present a method for constructing optimal fault-tolerant approximations of arbitrary unitary gates using an arbtrary discrete universal gate set. The method presented is numerical and scales exponentially with the number of gates used in the approximation, however, for the specific case of arbitrary single-qubit gates and the fault-tolerant gates permitted by the 7-qubit Steane code, it is shown that the longest practical gates sequences can be found. We also analyse the practicality of the fault-tolerant approximations of the phase rotation gates used in Shor’s algorithm and find that simple non-fault-tolerant phase rotations are more robust for realistic error rates. A general scaling law of how rapidly these fault-tolerant approximations converge to arbitrary single-qubit gates is also determined. In large-scale quantum computation, every qubit of data is encoded across multiple physical qubits to form a logical qubit permitting quantum error correction
Recent years have enjoyed a strong interest in exploring properties and applications of random quantum circuits. In this work, we explore the ensemble of t -doped Clifford circuits on n qubits, consisting of Clifford circuits interspersed with t single-qubit non-Clifford gates. We establish rigorous convergence bounds toward unitary k -designs, revealing the intrinsic cost in terms of non-Clifford resources in various flavors. First, we analyze the k th order frame potential, which quantifies how well the ensemble of doped Clifford circuits is spread within the unitary group. We prove that a quadratic doping level, t = Θ ~ ( k 2 ) , is both necessary and sufficient to approximate the frame potential of the full unitary group. As a consequence, we refine existing upper bounds on the convergence of the ensemble toward state k -designs. Second, we derive tight bounds on the convergence of t -doped Clifford circuits toward relative-error k -designs, showing that t = Θ ~ ( n k ) is both necessary and sufficient for the ensemble to form a relative ε -approximate k -design. Similarly, t = Θ ~ ( n ) is required to generate pseudorandom unitaries. All these results highlight that generating random unitaries is extremely costly in terms of non-Clifford resources, and that such ensembles fundamentally lie beyond the classical simulability barrier. Additionally, we analyze high-order doped-Clifford-Weingarten functions to derive analytic expressions for the twirling operator over the ensemble of random doped Clifford circuits, and we establish their asymptotic behavior in relevant regimes.