Random quantum circuits take an input quantum state and randomize it. This is a task with a growing number of identified uses in quantum information processing. We suggest a scheme to implement random circuits in a weighted graph state. The input state is entangled with the weighted graph state and a random circuit is implemented when the experimenter performs local measurements in one fixed basis only. The scheme uses no classical random numbers and is a new and natural application of weighted graph states
The Hidden Weighted Bit function plays an important role in the study of classical models of computation. A common belief is that this function is exponentially hard for the implementation by reversible ancilla-free circuits, even though introducing a small number of ancillae allows a very efficient implementation. In this paper, we refute the exponential hardness conjecture by developing a polynomial-size reversible ancilla-free circuit computing the Hidden Weighted Bit function. Our circuit has size $O(n^{6.42})$, where $n$ is the number of input bits. We also show that the Hidden Weighted B…
We show that low-depth random quantum circuits can be efficiently simulated by a quantum teleportation-inspired algorithm. By using logical qubits to redirect and teleport the quantum information in quantum circuits, the original circuits can be renormalized to new circuits with a smaller number of logical qubits. We demonstrate the algorithm to simulate several random quantum circuits, including 1D-chain 1000-qubit 42-depth, 2D-grid 125*8-qubit 42-depth and 2D-Bristlecone 72-qubit 32-depth circuits. Our results present a memory-efficient method with a clear physical picture to simulate low-de…
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.
We explicitly construct a quantum circuit which exactly generates random three-qubit states. The optimal circuit consists of three CNOT gates and fifteen single qubit elementary rotations, parametrized by fourteen independent angles. The explicit distribution of these angles is derived, showing that the joint distribution is a product of independent distributions of individual angles apart from four angles.
Sampling from probability distributions of quantum circuits is a fundamentally and practically important task which can be used to demonstrate quantum supremacy using noisy intermediate-scale quantum devices. In the present work, we examine classical simulability of sampling from the output photon-number distribution of linear-optical circuits composed of random beam splitters with equally distributed squeezed vacuum states and single-photon states input. We provide efficient classical algorithms to simulate linear-optical random circuits and show that the algorithms' error is exponentially sm…
Efficient implementation of randomized quantum algorithms with dynamic circuits
Mitsubishi Chemical Corporation（三菱化学公司）；Quantum Computing Center, Keio University（庆应义塾大学量子计算中心）；IBM Quantum, IBM Research – Tokyo（IBM量子，IBM研究-东京）；Department of Applied Physics and Physico-Informatics, Keio University（庆应义塾大学应用物理与物理信息系）
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 pseudo-random circuits, with the goal of identifying relevant trade-offs 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…
Given a universal gate set on two qubits, it is well known that applying random gates from the set to random pairs of qubits will eventually yield an approximately Haar-distributed unitary. However, this requires exponential time. We show that random circuits of only polynomial length will approximate the first and second moments of the Haar distribution, thus forming approximate 1- and 2-designs. Previous constructions required longer circuits and worked only for specific gate sets. As a corollary of our main result, we also improve previous bounds on the convergence rate of random walks on t…
In [A.W. Harrow and R.A. Low, Commun. Math. Phys. 291, 257-302 (2009)], it was shown that a quantum circuit composed of random 2-qubit gates converges to an approximate quantum 2-design in polynomial time. We point out and correct a flaw in one of the paper's main arguments. Our alternative argument highlights the role played by transpositions induced by the random gates in achieving convergence.
We explore the implementation of pseudo-random single-qubit rotations and multi-qubit pseudo-random circuits constructed only from Clifford gates and the T-gate, a phase rotation of 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 require significantly fewer gates than the construction of arbitrary single-qubit rotations. For Clifford plus T pseudo-random circuits we find an exponential c…
We test the principle of majorization [J. I. Latorre and M. A. Martin-Delgado, Phys. Rev. A 66, 022305 (2002)] in random circuits. Three classes of circuits were considered: (i) universal, (ii) classically simulatable, and (iii) neither universal nor classically simulatable. The studied families are: {CNOT, H, T}, {CNOT, H, NOT}, {CNOT, H, S} (Clifford), matchgates, and IQP (instantaneous quantum polynomial-time). We verified that all the families of circuits satisfy on average the principle of decreasing majorization. In most cases the asymptotic state (number of gates going to infinity) beha…
A major advantage in using Szegedy's formalism over discrete-time and continuous-time quantum walks lies in its ability to define a unitary quantum walk on directed and weighted graphs. In this paper, we present a general scheme to construct efficient quantum circuits for Szegedy quantum walks that correspond to classical Markov chains possessing transformational symmetry in the columns of the transition matrix. In particular, the transformational symmetry criteria do not necessarily depend on the sparsity of the transition matrix, so this scheme can be applied to non-sparse Markov chains. Two…
We consider a class of random quantum circuits where at each step a gate from a universal set is applied to a random pair of qubits, and determine how quickly averages of arbitrary finite-degree polynomials in the matrix elements of the resulting unitary converge to Haar measure averages. This is accomplished by establishing an exact mapping between the superoperator that describes t-order moments on n qubits and a multilevel SU(4^t) Lipkin-Meshkov-Glick Hamiltonian. For arbitrary fixed t, we find that the spectral gap scales as 1/n in the thermodynamic limit. Our results imply that random qua…
Randomized benchmarking is a useful scheme for evaluation the average fidelity of a noisy quantum circuit. However, it is insensitive to the unitary error. Here, we propose a method of randomized benchmarking in which a unitary t-design is applied and by which the unitary error estimation can be converted to analysis of pseudo-randomness on a set of unitary operators. We give a bound on the number of randomized benchmarking sequences, when performing a unitary t-design on n-qubit d-dimensional system. By applying local random unitary operators, a decomposition of a unitary t-design, the bound …
We give a general method of construting quantum circuit for random \QTR{it}{satisfiability} (SAT) problems with the basic logic gates such as multi-qubit controlled-NOT and NOT gates. The sizes of these circuits are almost the same as the sizes of the SAT formulas. Further, a parallelization scheme is described to solve random SAT problems efficiently through these quantum circuits in \QTR{it}{nuclear magnetic resonance} (NMR) ensemble quantum computing. This scheme exploits truly mixed states as input states rather than pseudo-pure states, and combines with the topological nanture of the NMR …
The performance of quantum gates is often assessed using some form of randomized benchmarking. However, the existing methods become infeasible for more than approximately five qubits. Here we show how to use a simple and customizable class of circuits -- randomized mirror circuits -- to perform scalable, robust, and flexible randomized benchmarking of Clifford gates. We show that this technique approximately estimates the infidelity of an average many-qubit logic layer, and we use simulations of up to 225 qubits with physically realistic error rates in the range 0.1-1% to demonstrate its scala…
Scaling of variational quantum algorithms to large problem sizes requires efficient optimization of random parameterized quantum circuits. For such circuits with uncorrelated parameters, the presence of exponentially vanishing gradients in cost function landscapes is an obstacle to optimization by gradient descent methods. In this work, we prove that reducing the dimensionality of the parameter space by utilizing circuit modules containing spatially or temporally correlated gate layers can allow one to circumvent the vanishing gradient phenomenon. Examples are drawn from random separable circu…
Resource generation and dynamical complexities in open random quantum circuits
Harish-Chandra Research Institute（哈里什-钱德拉研究所）；Homi Bhabha National Institute（霍米·巴巴国立研究所）；Indian Institute of Technology (ISM) Dhanbad（印度理工学院（ISMT）丹巴德分校）
We study efficient generations of random diagonal-unitary matrices, an ensemble of unitary matrices diagonal in a given basis with randomly distributed phases for their eigenvalues. Despite the simple algebraic structure, they cannot be achieved by quantum circuits composed of a few-qubit diagonal gates. We introduce diagonal-unitary $t$-designs and present two quantum circuits that implement diagonal-unitary $2$-designs with the computational basis in $N$-qubit systems. One is composed of single-qubit diagonal gates and controlled-phase gates with randomized phases, which achieves an exact di…
Towards verifiable quantum advantage with random circuits: Observables that survive concentration
Freie Universität Berlin（柏林自由大学）；Google Quantum AI（谷歌量子人工智能）；Institute for Quantum Information and Matter, California Institute of Technology（加州理工学院量子信息与物质研究所）
Sample-efficient benchmarking of shallow all-to-all random quantum circuits
The College of William & Mary（威廉玛丽学院）；The University of Chicago（芝加哥大学）；Joint Center for Quantum Information and Computer Science, University of Maryland and NIST（马里兰大学和NIST联合量子信息与计算中心）；University of Waterloo（滑铁卢大学）
Entanglement is the defining characteristic of quantum mechanics. Bipartite entanglement is characterized by the von Neumann entropy. Entanglement is not just described by a number, however; it is also characterized by its level of complexity. The complexity of entanglement is at the root of the onset of quantum chaos, universal distribution of entanglement spectrum statistics, hardness of a disentangling algorithm and of the quantum machine learning of an unknown random circuit, and universal temporal entanglement fluctuations. In this paper, we numerically show how a crossover from a simple …
Quantum Random Walks, which have drawn much attention over the past few decades for their distinctly non-classical behavior, is a promising subfield within Quantum Computing. Theoretical framework and applications for these walks have seen many great mathematical advances, with experimental demonstrations now catching up. In this study, we examine the viability of implementing Coin Quantum Random Walks using a Quantum Adder based Shift Operator, with quantum circuit designs specifically for superconducting qubits. We focus on the strengths and weaknesses of these walks, particularly circuit de…
We study the distribution over measurement outcomes of noisy random quantum circuits in the low-fidelity regime. We show that, for local noise that is sufficiently weak and unital, correlations (measured by the linear cross-entropy benchmark) between the output distribution $p_{\text{noisy}}$ of a generic noisy circuit instance and the output distribution $p_{\text{ideal}}$ of the corresponding noiseless instance shrink exponentially with the expected number of gate-level errors, as $F=\text{exp}(-2sε\pm O(sε^2))$, where $ε$ is the probability of error per circuit location and $s$ is the numbe…
Decoupling has become a central concept in quantum information theory with applications including proving coding theorems, randomness extraction and the study of conditions for reaching thermal equilibrium. However, our understanding of the dynamics that lead to decoupling is limited. In fact, the only families of transformations that are known to lead to decoupling are (approximate) unitary two-designs, i.e., measures over the unitary group which behave like the Haar measure as far as the first two moments are concerned. Such families include for example random quantum circuits with O(n^2) ga…
We prove that local random quantum circuits acting on n qubits composed of O(t^{10} n^2) many nearest neighbor two-qubit gates form an approximate unitary t-design. Previously it was unknown whether random quantum circuits were a t-design for any t > 3. The proof is based on an interplay of techniques from quantum many-body theory, representation theory, and the theory of Markov chains. In particular we employ a result of Nachtergaele for lower bounding the spectral gap of frustration-free quantum local Hamiltonians; a quasi-orthogonality property of permutation matrices; a result of Oliveira …
Pseudo-random operators consist of sets of operators that exhibit many of the important statistical features of uniformly distributed random operators. Such pseudo-random sets of operators are most useful whey they may be parameterized and generated on a quantum processor in a way that requires exponentially fewer resources than direct implementation of the uniformly random set. Efficient pseudo-random operators can overcome the exponential cost of random operators required for quantum communication tasks such as super-dense coding of quantum states and approximately secure quantum data-hiding…
The generation of certifiable randomness is the most fundamental information-theoretic task that meaningfully separates quantum devices from their classical counterparts. We propose a protocol for exponential certified randomness expansion using a single quantum device. The protocol calls for the device to implement a simple quantum circuit of constant depth on a 2D lattice of qubits. The output of the circuit can be verified classically in linear time, and is guaranteed to contain a polynomial number of certified random bits assuming that the device used to generate the output operated using …
The ability to efficiently simulate random quantum circuits using a classical computer is increasingly important for developing Noisy Intermediate-Scale Quantum devices. Here we present a tensor network states based algorithm specifically designed to compute amplitudes for random quantum circuits with arbitrary geometry. Singular value decomposition based compression together with a two-sided circuit evolution algorithm are used to further compress the resulting tensor network. To further accelerate the simulation, we also propose a heuristic algorithm to compute the optimal tensor contraction…
We present improved circuits intended for building a universal computer based on Random Pulse Computing (RPC) paradigm, a biologically-inspired way of computation in which variable is represented by a frequency of a Random Pulse Train (RPT) rather than a logic state. The RPC we mention here is also known as "stochastic unipolar computation" in newer literature. Unlike in previous art, where randomness is obtained from electronics noise or a pseudorandom shift register while processing circuitry is deterministic, in our approach both variable generation and signal processing rely on the random …
We consider the hardness of computing additive approximations to output probabilities of random quantum circuits. We consider three random circuit families, namely, Haar random, $p=1$ QAOA, and random IQP circuits. Our results are as follows. For Haar random circuits with $m$ gates, we improve on prior results by showing $\mathsf{coC_=P}$ hardness of average-case additive approximations to an imprecision of $2^{-O(m)}$. Efficient classical simulation of such problems would imply the collapse of the polynomial hierarchy. For constant depth circuits i.e., when $m=O(n)$, this linear scaling in th…
A continuous-time quantum walk on a graph evolves according to the unitary operator $e^{-iAt}$, where $A$ is the adjacency matrix of the graph. Perfect state transfer (PST) in a quantum walk is the transfer of a quantum state from one node of a graph to another node with $100\%$ fidelity. It can be shown that the adjacency matrix of a cubelike graph is a finite sum of tensor products of Pauli $X$ operators. We use this fact to construct an efficient quantum circuit for the quantum walk on cubelike graphs. In \cite{Cao2021, rishi2021(2)}, a characterization of integer weighted cubelike graphs i…
