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arXiv:quant-ph/0503210 · Physical Review A

Convergence conditions for random quantum circuits

Joseph Emerson, Etera R. Livine, Seth Lloyd

Published Dec 2, 2005• 94 citations
arXivHTMLPDF

Abstract

Efficient methods for generating pseudorandomly distributed unitary operators are needed for the practical application of Haar-distributed random operators in quantum communication and noise estimation protocols. We develop a theoretical framework for analyzing pseudorandom ensembles generated through a random circuit composition. We prove that the measure over random circuits converges exponentially (with increasing circuit length) to the uniform (Haar) measure on the unitary group, though the rate for uniform convergence must decrease exponentially with the number of qubits. We describe how the rate of convergence for test functions associated with specific randomization tasks leads to weaker convergence conditions that may allow efficient random circuit constructions.

DOI
10.1103/physreva.72.060302

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Efficient methods for generating pseudorandomly distributed unitary operators are needed for the practical application of Haar-distributed random operators in quantum communication and noise estimation protocols. We develop a theoretical framework for analyzing pseudorandom ensembles generated through a random circuit composition. We prove that the measure over random circuits converges exponentially (with increasing circuit length) to the uniform (Haar) measure on the unitary group, though the rate for uniform convergence must decrease exponentially with the number of qubits. We describe how the rate of convergence for test functions associated with specific randomization tasks leads to weaker convergence conditions that may allow efficient random circuit constructions.
Efficient methods for generating pseudo-randomly distributed unitary operators are needed for the practical application of Haar distributed random operators in quantum communication and noise estimation protocols. We develop a theoretical framework for analyzing pseudo-random ensembles generated through a random circuit composition. We prove that the measure over random circuits converges exponentially (with increasing circuit length) to the uniform (Haar) measure on the unitary group and describe how the rate of convergence may be calculated for specific applications.
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.
The applications of random quantum circuits range from quantum computing and quantum many-body systems to the physics of black holes. Many of these applications are related to the generation of quantum pseudorandomness: Random quantum circuits are known to approximate unitary t -designs. Unitary t -designs are probability distributions that mimic Haar randomness up to t th moments. In a seminal paper, Brandão, Harrow and Horodecki prove that random quantum circuits on qubits in a brickwork architecture of depth O(nt10.5) are approximate unitary t -designs. In this work, we revisit this argument, which lower bounds the spectral gap of moment operators for local random quantum circuits by Ω(n−1t−9.5) . We improve this lower bound to Ω(n−1t−4−o(1)) , where the o(1) term goes to 0 as t→∞ . A direct consequence of this scaling is that random quantum circuits generate approximate unitary t -designs in depth O(nt5+o(1)) . Our techniques involve Gao's quantum union bound and the unreasonable effectiveness of the Clifford group. As an auxiliary result, we prove fast convergence to the Haar measure for random Clifford unitaries interleaved with Haar random single qubit unitaries.
This thesis discusses the young fields of quantum pseudo-randomness and quantum learning algorithms. We present techniques for derandomising algorithms to decrease randomness resource requirements and improve efficiency. One key object in doing this is a k-design, which is a distribution on the unitary group whose kth moments match those of the unitarily invariant Haar measure. We show that for a natural model of a random quantum circuit, the distribution of random circuits quickly converges to a 2-design. We then present an efficient unitary k-design construction for any k, provided the number of qubits n satisfies k = O(n/log n). In doing this, we provide an efficient construction of a quantum tensor product expander, which is a generalisation of a quantum expander which in turn generalises classical expanders. We then discuss applications of k-designs. We show that they can be used to improve the efficiency of many existing algorithms and protocols and also find new applications to derandomising large deviation bounds. In particular, we show that many large deviation bound results for Haar random unitaries carry over to k-designs for k = poly(n). In the second part of the thesis, we present some learning and testing algorithms for the Clifford group. We find an optimal algorithm for identifying an unknown Clifford operation. We also give an algorithm to test if an unknown operation is close to a Clifford or far from every Clifford.
According to one of the fundamental axioms of quantum mechanics, unitary operators rule the evolution of any quantum system; it is thus of prominent importance to investigate the corresponding group and discover its properties. Random processes over the unitary group have indeed a wide range of applications in the context of quantum information; in particular, they are involved in the construction of peculiar distributions called unitary designs which mimic the uniform Haar measure by matching its moments and are thus deeply connected to the description of phenomena such as quantum tomography, equilibration, thermalization, encryption and scrambling. Previous approaches have shown that unitary designs are efficiently approximated by random quantum circuits with local unitary operators. Moreover, these circuits are used to characterize the precision of experimental implementations of unitary gates via randomized benchmarking protocols and to rapidly decouple a system from the environment. The main novel contribution of this work is to extend these mixing properties to a continuous-time framework, namely, Brownian motion over the unitary group induced by stochastic local Hamiltonians. In order to achieve these new results, on the one hand we move to a representation theoretic formulation and make use of its tools to establish the gap of local generators linked to the moments of the distribution induced by the diffusion process; from this we then derive an expression for the length of time it takes to ensure convergence toward the moments of the Haar measure. On the other hand we project the stochastic evolution onto a random walk on Pauli matrices and tie this description to the analogous one for random quantum circuits to achieve decoupling in a run time scaling almost linearly with respect to the system size. In addition to providing a unifying framework for random processes over the unitary group, we hence aim at presenting new mathematical results and techniques for quantum information. We furthermore discuss applications to black holes dynamics in the perspective of the information paradox. As an additional novel result not related to Brownian motion, we propose a randomized benchmarking protocol exploiting the symmetry of the gate whose accuracy has to be estimated, in order to overcome current shortcomings afflicting the known schemes. We again rely on representation theory to reduce the computational effort and the amount of employed quantum resources.
Pseudo‐random operators consist of sets of operators that exhibit important statistical features of uniformly distributed operators although their parametrization and generation requires exponentially fewer resources. Pseudo‐random operators can be applied for quantum communication tasks such as data‐hiding and super‐dense coding of quantum states, and enable efficient stochastic methods for noise estimation on prototype quantum processors. This paper summarizes published work by Emerson et al. [Science 302:2098 (2003)] demonstrating a random circuit method for the implementation of pseudo‐random unitary operators on a quantum processor and some more recent related results.