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arXiv:quant-ph/0512217

Efficient Simulation of Random Quantum States and Operators

Christoph Dankert

Published Dec 23, 2005• 39 citations
arXivHTMLPDF

Abstract

We investigate the generation of quantum states and unitary operations that are ``random'' in certain respects. We show how to use such states to estimate the average fidelity, an important measure in the study of implementations of quantum algorithms. We re-discover the result that the states of a maximal set of mutually-unbiased bases serve this purpose. An efficient circuit is presented that generates an arbitrary state out of such a set. Later on, we consider unitary operations that can be used to turn any quantum channel into a depolarizing channel. It was known before that the Clifford group serves this and a related purpose, and we show that these are actually the same. We also show that a small subset of the Clifford group is already sufficient to accomplish this. We conclude with an efficient construction of the elements of that subset. This thesis is based on joint work with Richard Cleve, Joseph Emerson, and Etera Livine.

DOI
10.48550/arxiv.quant-ph/0512217

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npj Quantum Information•Apr 26, 2016•91 citations
We investigate the generation of quantum states and unitary operations that are ``random'' in certain respects. We show how to use such states to estimate the average fidelity, an important measure in the study of implementations of quantum algorithms. We re-discover the result that the states of a maximal set of mutually-unbiased bases serve this purpose. An efficient circuit is presented that generates an arbitrary state out of such a set. Later on, we consider unitary operations that can be used to turn any quantum channel into a depolarizing channel. It was known before that the Clifford group serves this and a related purpose, and we show that these are actually the same. We also show that a small subset of the Clifford group is already sufficient to accomplish this. We conclude with an efficient construction of the elements of that subset. This thesis is based on joint work with Richard Cleve, Joseph Emerson, and Etera Livine.
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.
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.
Random Clifford operators have numerous applications in quantum computing, including randomized benchmarking, classical shadows, and quantum authentication. However, sampling and implementing uniformly random nnn-qubit Clifford incur near-quadratic complexity due to the size of Clifford group. We introduce a cryptographic way to overcome these barriers: trapdoored Clifford operator distributions whose samples are computationally indistinguishable from uniformly random Cliffords, yet implementing them can be much faster given the trapdoor. We construct a distribution of trapdoored Clifford operators whose elements can be sampled and implemented in near-linear time under a variant of the learning parity with noise assumption. Our constructions allow fast tableau action on Pauli labels for classical simulation, and also can be optimized to admit polylogarithmic-depth implementation. Along the way, we construct trapdoored matrices over finite fields that support efficient multiplication by both a matrix and its inverse, resolving an open question left by Vaikuntanathan and Zamir [SODA'26]. We use these constructions to obtain faster protocols based on random Cliffords. We also explore their applications to the worst-case to average-case reductions for matrix and Clifford problems including the iterated matrix multiplication and Clifford circuit synthesis. In particular, we show the hardness of batching Clifford circuits: synthesizing circuits that apply the same Clifford to multiple registers is at least as hard as worst-case matrix multiplication, even when synthesis succeeds on a small constant fraction of random Cliffords. This extends to approximate implementations by general quantum circuits.
The purpose of the research: to determine an efficient and error-resilient method for assessing the fidelity of singlequbit quantum operations, as well as to formulate and experimentally implement an algorithm for determining the average accuracy of single-qubit quantum transformations Research methods: the theory of noisy quantum operations, randomization of single-qubit quantum circuits composed of Clifford group transformations, the theory of unitary 2-design, experimental validation on a quantum computer based on ytterbium-171 trapped ions Research results: an algorithm for estimating the average fidelity of single-qubit operations and its statistical error is described; experimental validation confirmed the correctness of the exponential decay model of the probability of obtaining the target state during measurement as the depth of the quantum circuit increases; the resulting experimental average fidelity of the single-qubit quantum operation is 99.94% Scientific novelty: the application of the randomized benchmarking method to determine the average fidelity of singlequbit quantum operations in quantum computing, particularly for an ion-based quantum processor with ytterbium-171 ions.
Randomized benchmarking is a widely used experimental technique to characterize the average error of quantum operations. Benchmarking procedures that scale to enable characterization of nnn-qubit circuits rely on efficient procedures for manipulating those circuits and, as such, have been limited to subgroups of the Clifford group. However, universal quantum computers require additional, non-Clifford gates to approximate arbitrary unitary transformations. We define a scalable randomized benchmarking procedure over nnn-qubit unitary matrices that correspond to protected non-Clifford gates for a class of stabilizer codes. We present efficient methods for representing and composing group elements, sampling them uniformly, and synthesizing corresponding poly(n)\mathrm{poly}(n)poly(n)-sized circuits. The procedure provides experimental access to two independent parameters that together characterize the average gate fidelity of a group element.
Randomised benchmarking is a widely used experimental technique to characterise the average error of quantum operations. Benchmarking procedures that scale to enable the characterisation of n-qubit circuits rely on efficient procedures for manipulating those circuits and, as such, have been limited to subgroups of the Clifford group. However, universal quantum computers require additional, non-Clifford gates to approximate arbitrary unitary transformations. We define a scalable randomised benchmarking procedure over n-qubit unitary matrices that correspond to protected non-Clifford gates for a class of stabiliser codes. We present efficient methods for representing and composing group elements, sampling them uniformly and synthesising corresponding poly(n)-sized circuits. The procedure provides experimental access to two independent parameters that together characterise the average gate fidelity of a group element. A scalable procedure to determine the error of quantum operations has been developed by researchers in the United States. Andrew Cross and colleagues at the IBM T.J. Watson Research Center have expanded previously used protocols to a broader range of quantum circuits, thereby enabling error benchmarking for all common logical operations used in leading quantum computing protocols. The standard technique for error characterization in these devices is randomized benchmarking, which provides benefits such as insensitivity to measurement errors, but on the other hand only functions in a limited number of circumstances. In their theoretical study, the researchers demonstrate how randomized benchmarking approaches can be expanded to a much broader range of logical operations. The procedure can be readily implemented experimentally, and provides access to important experimental noise parameters in quantum computing.