Random quantum circuits are a central concept in quantum information theory with applications ranging from demonstrations of quantum computational advantage to descriptions of scrambling in strongly-interacting systems and black holes. The utility of random quantum circuits in these settings stems from their ability to rapidly generate quantum pseudo-randomness. In a seminal paper by Brandão, Harrow, and Horodecki, it was proven that the $t$-th moment operator of local random quantum circuits on $n$ qudits with local dimension $q$ has a spectral gap of at least $Ω(n^{-1}t^{-5-3.1/\log(q)})$, which implies that they are efficient constructions of approximate unitary designs. As a first result, we use Knabe bounds for the spectral gaps of frustration-free Hamiltonians to show that $1D$ random quantum circuits have a spectral gap scaling as $Ω(n^{-1})$, provided that $t$ is small compared to the local dimension: $t^2\leq O(q)$. This implies a (nearly) linear scaling of the circuit depth in the design order $t$. Our second result is an unconditional spectral gap bounded below by $Ω(n^{-1}\log^{-1}(n) t^{-α(q)})$ for random quantum circuits with all-to-all interactions. This improves both the $n$ and $t$ scaling in design depth for the non-local model. We show this by proving a recursion relation for the spectral gaps involving an auxiliary random walk. Lastly, we solve the smallest non-trivial case exactly and combine with numerics and Knabe bounds to improve the constants involved in the spectral gap for small values of $t$.
Exact spectral gaps of random one-dimensional quantum circuits
Los Alamos National Laboratory（洛斯阿拉莫斯国家实验室）；Donostia International Physics Center（圣塞巴斯蒂安国际物理中心）；Department of Applied Physics, Gipuzkoa School of Engineering, University of the Basque Country (UPV/EHU)（巴斯克国立大学吉普斯夸工程学院应用物理系）；Theoretical Division, Los Alamos National Laboratory（洛斯阿拉莫斯国家理论部）；Quantum Science Center, Oak Ridge, TN 37931, USA（橡树岭量子科学中心）
Finite-size criteria have emerged as an effective tool for deriving spectral gaps in higher-dimensional frustration-free quantum spin systems. We quantitatively improve the existing finite-size criteria by introducing a novel subsystem weighting scheme. The approach applies to Euclidean lattices of any dimension, the honeycomb lattice, and the triangular lattice.
We study Landau-Zener-Stückelberg (LZS) interferometry in a cQED architecture under effects of dissipation. To be specific, we consider a superconducting qubit driven by a dc+ac signal and coupled to a transmission line resonator, but our results are valid for general qubit-resonators devices. To take the environment into account, we assume that the resonator is coupled to an ohmic quantum bath. The Floquet-Born-Markov master equation is numerically solved to obtain the dynamics of the system for arbitrary amplitude of the drive and different time scales. We unveil important differences in the…
In this work, we propose a generalization of the current most widely used quantum computing hardware metric known as the quantum volume. The quantum volume specifies a family of random test circuits defined such that the logical circuit depth is equal to the total number of qubits used in the computation. However, such square circuit shapes do not directly relate to many specific applications for which one may wish to use a quantum computer. Based on surveying available resource estimates for known quantum algorithms, we generalize the quantum volume to a handful of representative circuit shap…
We present improved quantum circuits for elliptic curve scalar multiplication, the most costly component in Shor's algorithm to compute discrete logarithms in elliptic curve groups. We optimize low-level components such as reversible integer and modular arithmetic through windowing techniques and more adaptive placement of uncomputing steps, and improve over previous quantum circuits for modular inversion by reformulating the binary Euclidean algorithm. Overall, we obtain an affine Weierstrass point addition circuit that has lower depth and uses fewer $T$ gates than previous circuits. While pr…
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 …
Bravyi and Gosset recently gave classical simulation algorithms for quantum circuits dominated by Clifford operations. These algorithms scale exponentially with the number of T-gate in the circuit, but polynomially in the number of qubits and Clifford operations. Here we extend their algorithm to qudits of odd prime dimensions. We generalize their approximate stabilizer rank method for weak simulation to qudits and obtain the scaling of the approximate stabilizer rank with the number of single-qudit magic states. We also relate the canonical form of qudit stabilizer states to Gauss sum evaluat…
We benchmark the efficacy of several novel orthogonal, symmetric, dilation-3 wavelets, derived from a unitary circuit based construction, towards image compression. The performance of these wavelets is compared across several photo databases against the CDF-9/7 wavelets in terms of the minimum number of non-zero wavelet coefficients needed to obtain a specified image quality, as measured by the multi-scale structural similarity index (MS-SSIM). The new wavelets are found to consistently offer better compression efficiency than the CDF-9/7 wavelets across a broad range of image resolutions and …
Improving the simulation of quantum circuits on classical computers is important for understanding quantum advantage and increasing development speed. In this paper, we explore a new way to express stabilizer states and further improve the speed of simulating stabilizer circuits with a current existing approach. First, we discover a unique and elegant canonical form for stabilizer states based on graph states to better represent stabilizer states and show how to efficiently simplify stabilizer states to canonical form. Second, we develop an improved algorithm for graph state stabilizer simulat…
Quantum measurement is important to quantum computing as it extracts the outcome of the circuit at the end of the computation. Previously, all measurements have to be done at the end of the circuit. Otherwise, it will incur significant errors. But it is not the case now. Recently IBM started supporting dynamic circuits through hardware (instead of software by simulator). With mid-circuit hardware measurement, we can improve circuit efficacy and fidelity from three aspects: (a) reduced qubit usage, (b) reduced swap insertion, and (c) improved fidelity. We demonstrate this using real-world appli…
Adiabatic quantum computing is a framework for quantum computing that is superficially very different to the standard circuit model. However, it can be shown that the two models are computationally equivalent. The key to the proof is a mapping of a quantum circuit to an an adiabatic evolution, and then showing that the minimum spectral gap of the adiabatic Hamiltonian is at least inverse polynomial in the number of computational steps $L$. In this paper we provide two simplified proofs that the gap is inverse polynomial. Both proofs result in the same lower bound for the minimum gap, which for…
Integer arithmetic is the underpinning of many quantum algorithms, with applications ranging from Shor's algorithm over HHL for matrix inversion to Hamiltonian simulation algorithms. A basic objective is to keep the required resources to implement arithmetic as low as possible. This applies in particular to the number of qubits required in the implementation as for the foreseeable future this number is expected to be small. We present a reversible circuit for integer multiplication that is inspired by Karatsuba's recursive method. The main improvement over circuits that have been previously re…
We show that the cost of strong simulation of quantum circuits using $t$ $T$ gate magic states exhibits non-trivial reductions on its upper bound for $t=1$, $t=2$, $t=3$, and $t=6$ with odd-prime-qudits. This agrees with previous numerical bounds found for qubits. We define simulation cost by the number of terms that require Gaussian elimination of a $t \times t$ matrix and so capture the cost of simulation methods that proceed by computing stabilizer inner products or evaluating quadratic Gauss sums. Prior numerical searchs for qubits were unable to converge beyond $t=7$. We effectively incre…
Present-day, noisy, small or intermediate-scale quantum processors---although far from fault-tolerant---support the execution of heuristic quantum algorithms, which might enable a quantum advantage, for example, when applied to combinatorial optimization problems. On small-scale quantum processors, validations of such algorithms serve as important technology demonstrators. We implement the quantum approximate optimization algorithm (QAOA) on our hardware platform, consisting of two superconducting transmon qubits and one parametrically modulated coupler. We solve small instances of the NP-comp…
Improved Strategies for Fermionic Quantum Simulation with Global Interactions
Institute of Materials Research, German Aerospace Center (DLR)（德国航空航天中心材料研究所）；Institute for Applied Physics, Technical University of Darmstadt（达姆施塔特工业大学应用物理研究所）
The Gottesman-Knill theorem asserts that a quantum circuit composed of Clifford gates can be efficiently simulated on a classical computer. Here we revisit this theorem and extend it to quantum circuits composed of Clifford and T gates, where T is the single-qubit 45-degree phase shift. We assume that the circuit outputs a bit string x obtained by measuring some subset of w qubits. Two simulation tasks are considered: (1) computing the probability of a given output x, and (2) sampling x from the output probability distribution. It is shown that these tasks can be solved on a classical computer…
The progress in building quantum computers to execute quantum algorithms has recently been remarkable. Grover's search algorithm in a binary quantum system provides considerable speed-up over classical paradigm. Further, Grover's algorithm can be extended to a $d$-ary (qudit) quantum system for utilizing the advantage of larger state space, which helps to reduce the run-time of the algorithm as compared to the traditional binary quantum systems. In a qudit quantum system, an $n$-qudit Toffoli gate plays a significant role in the accurate implementation of Grover's algorithm. In this article, a…
Optimal and improved gate decompositions for accelerated classical simulation of near-Gaussian fermionic circuits
Phasecraft Ltd（Phasecraft有限公司）；Department of Mathematics, School of Computation, Information and Technology, Technical University of Munich（慕尼黑工业大学计算、信息与技术学院数学系）；Munich Center for Quantum Science and Technology（慕尼黑量子科学与技术中心）
Computing band gaps of periodic materials via sample-based quantum diagonalization
IBM Research（IBM研究院）；Instituto de Física, Universidade Federal Fluminense（弗鲁米嫩塞联邦大学物理研究所）；Centro Brasileiro de Pesquisas Físicas（巴西物理研究中心）；IBM Research Europe（IBM欧洲研究院）；National Center for Competence in Research-Catalysis (NCCR-Catalysis)（国家卓越研究中心-催化）；IBM Quantum, IBM T.J. Watson Research Center（IBM量子，IBM T.J.沃森研究中心）
Minimizing CNOT-count in quantum circuit of the extended Shor's algorithm for ECDLP
Institute of Information Engineering, CAS（中国科学院信息工程研究所）；School of Cyber Security, University of Chinese Academy of Sciences（中国科学院大学网络安全学院）；Institute of Information Engineering, Chinese Academy of Sciences（中国科学院信息工程研究所）
Bounding the cost of classically simulating the outcomes of universal quantum circuits to additive error $δ$ is often called weak simulation and is a direct way to determine when they confer a quantum advantage. Weak simulation of the $T$+Clifford gateset is $BQP$-complete and is expected to scale exponentially with the number $t$ of $T$ gates. We constructively tighten the upper bound on the worst-case $L_1$ norm sampling cost to next order in $t$ from $\mathcal O(ξ^t δ^{-2})$ if $δ^2 \gg ξ^{-t}$ to $\mathcal O((ξ^t{-}t) δ^{-2} )$ if $δ^2 \gg (ξ^t -t)^{-1}$, where $ξ^t = 2^{\sim 0.228 t}$ is …
Emerging quantum algorithms that process data require that classical input data be represented as a quantum state. These data-processing algorithms often follow the gate model of quantum computing--which requires qubits to be initialized to a basis state, typically $\lvert 0 \rangle$--and thus often employ state generation circuits to transform the initialized basis state to a data-representation state. There are many ways to encode classical data in a qubit, and the oft-applied approach of basis encoding does not allow optimization to the extent that other variants do. In this work, we thus c…
Qutrit (or ternary) structures arise naturally in many quantum systems, particularly in certain non-abelian anyon systems. We present efficient circuits for ternary reversible and quantum arithmetics. Our main result is the derivation of circuits for two families of ternary quantum adders, namely ripple carry adders and carry look-ahead adders. The main difference to the binary case is the more complicated form of the ternary carry, which leads to higher resource counts for implementations over a universal ternary gate set. Our ternary ripple adder circuit has a circuit depth of $O(n)$ and use…
Improving the accuracy of circuit quantization using the electromagnetic properties of superconductors
Seoul National University（首尔大学）；Korea Research Institute of Standards and Science（韩国标准科学研究院）；Quantum Devices Labs, Korea Advanced Nano Fab Center（韩国先进纳米制造中心量子器件实验室）；Quantum Design Korea（量子设计韩国公司）；SKKU Adavanced Institute of Nanotechnology, Sungkyunkwan University（成均馆大学SKKU纳米技术高级研究所）
In some quantum algorithms, arithmetic operations are of utmost importance for resource estimation. In binary quantum systems, some efficient implementation of arithmetic operations like, addition/subtraction, multiplication/division, square root, exponential and arcsine etc. have been realized, where resources are reported as a number of Toffoli gates or T gates with ancilla. Recently it has been demonstrated that intermediate qutrits can be used in place of ancilla, allowing us to operate efficiently in the ancilla-free frontier zone. In this article, we have incorporated intermediate qutrit…
