本文聚焦于三种关键矩阵操作的量子算法：Hadamard（Schur）积、Kronecker（张量）积以及初等列变换。通过设计特定的酉变换和辅助量子测量，提出了带有电路图的高效量子方案。它们的计算深度分别为：Kronecker积为O(1)；Hadamard积为O(max(m,n))（与矩阵维度相关）；(2^n X 2^m)矩阵的初等列变换为O(m)（仅取决于列数）。值得注意的是，与通过矩阵转置和行变换实现的传统列变换相比，该方案减少了计算步骤和量子门使用，降低了量子计算能耗。
英文摘要：
This paper focuses on quantum algorithms for three key matrix operations: Hadamard (Schur) product, Kronecker (tensor) product, and elementary column transformations each. By designing specific unitary transformations and auxiliary quantum measurement, efficient quantum schemes with circuit diagrams are proposed. Their computational depths are: O(1) for Kronecker product; O(max(m,n)) for Hadamard product (linked to matrix dimensions); and O(m) for elementary column transformations of (2^n X 2^m) matrices (dependent only on column count).Notably, compared to traditional column transformation via matrix transposition and row transformations, this scheme reduces computation steps and quantum gate usage, lowering quantum computing energy costs.
We propose the generalized controlled X (GCX) gate as the two-qudit elementary gate, and based on Cartan decomposition, we also give the one-qudit elementary gates. Then we discuss the physical implementation of these elementary gates and show that it is feasible with current technology. With these elementary gates many important qudit quantum gates can be synthesized conveniently. We provide efficient methods for the synthesis of various kinds of controlled qudit gates and greatly simplify the synthesis of existing generic multi-valued quantum circuits. Moreover, we generalize the quantum Sha…
In this article the elementary gates for ternary quantum logic circuit are studied. We propose the ternary controlled X (TCX) gate or ternary controlled Z (TCZ) gate as two-qutrit elementary gate, which is universal when assisted by arbitrary one-qutrit gates. It is primitive, efficient and easy to implement. Based on Cartan decomposition, we also give the one-qutrit elementary gates. Then the synthesis of some important ternary gates is investigated and the scheme of physical implementation for these ternary gates is discussed. Finally we extend these elementary gates to more general qudit ca…
We consider a generic elementary gate sequence which is needed to implement a general quantum gate acting on n qubits -- a unitary transformation with 4^n degrees of freedom. For synthesizing the gate sequence, a method based on the so-called cosine-sine matrix decomposition is presented. The result is optimal in the number of elementary one-qubit gates, 4^n, and scales more favorably than the previously reported decompositions requiring 4^n-2^n+1 controlled NOT gates.
We construct a minimal robust controlled-NOT gate with an Ising-type interaction by which elementary two-qubit gates are implemented. It is robust against inaccuracy of the coupling strength and the obtained quantum circuits are constructed with the minimal number (N=3) of elementary two-qubit gates and several one-qubit gates. It is noteworthy that all the robust circuits can be mapped to one-qubit circuits robust against a pulse length error. We also prove that a minimal robust SWAP gate cannot be constructed with N=3, but requires N=6 elementary two-qubit gates.
Quantum circuits currently constitute a dominant model for quantum computation. Our work addresses the problem of constructing quantum circuits to implement an arbitrary given quantum computation, in the special case of two qubits. We pursue circuits without ancilla qubits and as small a number of elementary quantum gates as possible. Our lower bound for worst-case optimal two-qubit circuits calls for at least 17 gates: 15 one-qubit rotations and 2 CNOTs. To this end, we constructively prove a worst-case upper bound of 23 elementary gates, of which at most 4 (CNOT) entail multi-qubit interacti…
Quantum noise or decoherence is a major factor impacting the performance of quantum technologies. On the qubit, an important quantum noise, often relevant in practice, is the thermal noise or generalized amplitude damping noise, describing the interaction with a thermal bath at an arbitrary temperature. A qubit thermal noise however cannot be modeled nor directly simulated with a few elementary Pauli operators, but instead requires specific operators. Our main goal here is to construct a circuit model for simulating the thermal noise from standard elementary qubit operators. Starting from a co…
We describe a simple algorithm for sampling $n$-qubit Clifford operators uniformly at random. The algorithm outputs the Clifford operators in the form of quantum circuits with at most $5n + 2n^2$ elementary gates and a maximum depth of $\mathcal{O}(n\log n)$ on fully connected topologies. The circuit can be output in a streaming fashion as the algorithm proceeds, and different parts of the circuit can be generated in parallel. The algorithm has an $\mathcal{O}(n^2)$ time complexity, which matches the current state of the art. The main advantage of the proposed algorithm, however, lies in its s…
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.
The complexity of a quantum gate, defined as the minimal number of elementary gates to build it, is an important concept in quantum information and computation. It is shown recently that the complexity of quantum gates built from random quantum circuits almost surely grows linearly with the number of building blocks. In this article, we provide two short proofs of this fact. We also discuss a discrete version of quantum circuit complexity growth.
Each platonic solid defines a single-qubit positive operator valued measure (POVM) by interpreting its vertices as points on the Bloch sphere. We construct simple circuits for implementing this kind of measurements and other simple types of symmetric POVMs on one qubit. Each implementation consists of a discrete Fourier transform and some elementary quantum operations followed by an orthogonal measurement in the computational basis.
In order to establish the computational equivalence between quantum Turing machines (QTMs) and quantum circuit families (QCFs) using Yao's quantum circuit simulation of QTMs, we previously introduced the class of uniform QCFs based on an infinite set of elementary gates, which has been shown to be computationally equivalent to the polynomial-time QTMs (with appropriate restriction of amplitudes) up to bounded error simulation. This result implies that the complexity class BQP introduced by Bernstein and Vazirani for QTMs equals its counterpart for uniform QCFs. However, the complexity classes …
We introduce transformation matrix connecting sets of the displaced states with different displacement amplitudes. Arbitrary pure one-mode state can be represented in new basis of the displaced number (Fock) states ( representation) by multiplying the transposed transformation matrix on a column vector of initial state. Analytical expressions of the representation of superposition of vacuum and single photon and two-mode squeezed vacuum (TMSV) are obtained. On the basis of the developed mathematical formalism, we consider the mechanism of interaction between qubits which is based on their disp…
In this work, we provide an overview of circuits for quantum computing. We introduce gates used in quantum computation and then present resource cost measurements used to evaluate circuits made from these gates. We then illustrate how the gates shown are then combined into quantum circuits for basic arithmetic functions. Architectures for addition, subtraction, multiplication, and division are shown. We demonstrate how to calculate the resource costs of quantum circuits. We conclude this overview with by illustrating an application of the elementary quantum circuits for the image rotation oper…
We try to minimize the number of qubits needed to factor an integer of n bits using Shor's algorithm on a quantum computer. We introduce a circuit which uses 2n+3 qubits and O(n^3 lg(n)) elementary quantum gates in a depth of O(n^3) to implement the factorization algorithm. The circuit is computable in polynomial time on a classical computer and is completely general as it does not rely on any property of the number to be factored. Keywords: Factorization, quantum circuits, modular arithmetics
We suggest an efficient scheme for quantum computation with linear optical elements utilizing "linked" photon states. The linked states are designed according to the particular quantum circuit one wishes to process. Once a linked-state has been successfully prepared, the computation is pursued deterministically by a sequence of teleportation steps. The present scheme enables a significant reduction of the average number of elementary gates per logical gate to about 20-30 CZ_{9/16} gates.
Arbitrary exponentially large unitaries cannot be implemented efficiently by quantum circuits. However, we show that quantum circuits can efficiently implement any unitary provided it has at most polynomially many nonzero entries in any row or column, and these entries are efficiently computable. One can formulate a model of computation based on the composition of sparse unitaries which includes the quantum Turing machine model, the quantum circuit model, anyonic models, permutational quantum computation, and discrete time quantum walks as special cases. Thus we obtain a simple unified proof t…
In general, a quantum circuit is constructed with elementary gates, such as one-qubit gates and CNOT gates. It is possible, however, to speed up the execution time of a given circuit by merging those elementary gates together into larger modules, such that the desired unitary matrix expressing the algorithm is directly implemented. We demonstrate this by taking the two-qubit Grover's algorithm implemented in NMR quantum computation, whose pseudopure state is generated by cyclic permutations of the state populations. This is the first exact time-optimal solution, to our knowledge, obtained for …
The design and optimization of quantum circuits is central to quantum computation. This paper presents new algorithms for compiling arbitrary 2^n x 2^n unitary matrices into efficient circuits of (n-1)-controlled single-qubit and (n-1)-controlled-NOT gates. We first present a general algebraic optimization technique, which we call the Palindrome Transform, that can be used to minimize the number of self-inverting gates in quantum circuits consisting of concatenations of palindromic subcircuits. For a fixed column ordering of two-level decomposition, we then give an numerative algorithm for min…
In this paper, we present a general quantum computation compiler, which maps any given quantum algorithm to a quantum circuit consisting a sequential set of elementary quantum logic gates based on recursive cosine-sine decomposition. The resulting quantum circuit diagram is provided by directly linking the package output written in LaTex to Qcircuit.tex $<$http://www.cquic.org/Qcircuit$>$. We illustrate the use of the Qcompiler package through various examples with full details of the derived quantum circuits. Besides its generality and simplicity, Qcompiler produces quantum circuits which ref…
How to implement quantum oracle with limited resources raises concerns these days. We design two ancilla-adjustable and efficient algorithms to synthesize SAT-oracle, the key component in solving SAT problems. The previous work takes 2m-1 ancillary qubits and O(m) elementary gates to synthesize an m clauses oracle. The first algorithm reduces the number of ancillary qubits to 2\sqrt{m}, with at most an eightfold increase in circuit size. The number of ancillary qubits can be further reduced to 3 with a quadratic increase in circuit size. The second algorithm aims to reduce the circuit depth. B…
Quantum computers hold unprecedented potentials for machine learning applications. Here, we prove that physical quantum circuits are PAC (probably approximately correct) learnable on a quantum computer via empirical risk minimization: to learn a parametric quantum circuit with at most $n^c$ gates and each gate acting on a constant number of qubits, the sample complexity is bounded by $\tilde{O}(n^{c+1})$. In particular, we explicitly construct a family of variational quantum circuits with $O(n^{c+1})$ elementary gates arranged in a fixed pattern, which can represent all physical quantum circui…
Quantum circuit optimization using deep reinforcement learning: Applications across multiple gate sets
Hanyang University（汉阳大学）；Hanyang Institute for Quantum Science and Quantum Technology, Hanyang University（汉阳大学量子科学与量子技术研究所）；Research Institute for Natural Sciences, Hanyang University（汉阳大学自然科学研究院）
We consider continuous quantum measurement of a superconducting qubit in the circuit QED setup with a moderate bandwidth of the measurement resonator, i.e., when the "bad cavity" limit is not applicable. The goal is a simple description of the quantum evolution due to measurement, i.e., the measurement back-action. Extending the quantum Bayesian approach previously developed for the "bad cavity" regime, we show that the evolution equations remain the same, but now they should be applied to the entangled qubit-resonator state, instead of the qubit state alone. The derivation uses only elementar…
The exponential speed up of quantum algorithms and the fundamental limits of current CMOS process for future design technology have directed attentions toward quantum circuits. In this paper, the matrix specification of a broad category of quantum circuits, i.e. CNOT-based circuits, are investigated. We prove that the matrix elements of CNOT-based circuits can only be zeros or ones. In addition, the columns or rows of such a matrix have exactly one element with the value of 1. Furthermore, we show that these specifications can be used to synthesize CNOT-based quantum circuits. In other words, …
Quantum circuit compilation with constant overhead
Google Quantum AI（谷歌量子人工智能）；Stanford University（斯坦福大学）；MIT（麻省理工学院）；Institute for Quantum Computing, University of Waterloo and Perimeter Institute for Theoretical Physics（滑铁卢大学量子计算研究所与佩里尔理论物理研究所）
We propose an implementation of the algorithm for the fast Fourier transform (FFT) as a quantum circuit consisting of a combination of some quantum gates. In our implementation, a data sequence is expressed by a tensor product of vector spaces. Namely, our FFT is defined as a transformation of the tensor product of quantum states. It is essentially different from the so-called quantum Fourier transform (QFT) defined to be a linear transformation of the amplitudes for the superposition of quantum states. The quantum circuit for the FFT consists of several circuits for elementary arithmetic oper…
We study the scaling of the convergence of several statistical properties of a recently introduced random unitary circuit ensemble towards their limits given by the circular unitary ensemble (CUE). Our study includes the full distribution of the absolute square of a matrix element, moments of that distribution up to order eight, as well as correlators containing up to 16 matrix elements in a given column of the unitary matrices. Our numerical scaling analysis shows that all of these quantities can be reproduced efficiently, with a number of random gates which scales at most as $n_q\log (n_q/ε)…
Recently, it is shown that quantum computers can be used for obtaining certain information about the solution of a linear system Ax=b exponentially faster than what is possible with classical computation. Here we first review some key aspects of the algorithm from the standpoint of finding its efficient quantum circuit implementation using only elementary quantum operations, which is important for determining the potential usefulness of the algorithm in practical settings. Then we present a small-scale quantum circuit that solves a 2x2 linear system. The quantum circuit uses only 4 qubits, imp…
We develop a unitary dependence theory to characterize the behaviors of quantum circuits and states in terms of how quantum gates manipulate qubits and determine their measurement probabilities. A qubit has dependence on a 1-qubit unitary gate if its measurement probabilities depend on the parameters of the gate. A 1-qubit unitary creates such a dependence onto the target qubit, and a CNOT gate copies all the dependences from the control qubit to the target qubit. The complete dependence picture of the output state details the connections qubits may have when being manipulated or measured. Com…
Recently, the dynamics simulation of light-harvesting complexes as an open quantum system, in the weak and strong coupling regimes, has received much attention. In this paper, we investigate a digital quantum simulation approach of the Fenna-Matthews-Olson (FMO) photosynthetic pigment-protein complex surrounded with a Markovian bath, i.e., memoryless, based on a nuclear magnetic resonance (NMR) quantum computer. For this purpose, we apply the decoupling(recoupling) method, which is turn off(on) the couplings and also Solovay-Kitaev techniques to decompose Hamiltonian and Lindbladians into effi…
We investigate the counterparts of random walk in universal quantum computing and their implementation using standard quantum circuits. Quantum walk have been recently well investigated for traversing graphs with certain oracles. We focus our study on traversing a 1-D graph, namely a circle, and show how to implement discrete circular quantum walk in quantum circuits built with universal CNOT and single quit gates. We review elementary quantum gates and circuit decomposition and propose a a generalized version of the all CNOT based quantum discrete circular walk. We simulated these circuits on…
In order to demonstrate non-trivial quantum computations experimentally, such as the synthesis of arbitrary entangled states, it will be useful to understand how to decompose a desired quantum computation into the shortest possible sequence of one-qubit and two-qubit gates. We contribute to this effort by providing a method to construct an optimal quantum circuit for a general two-qubit gate that requires at most 3 CNOT gates and 15 elementary one-qubit gates. Moreover, if the desired two-qubit gate corresponds to a purely real unitary transformation, we provide a construction that requires at…
We design a circuit structure with linear depth to implement an $n$-qubit Toffoli gate. The proposed construction uses a quadratic-size circuit consists of elementary 2-qubit controlled-rotation gates around the x axis and uses no ancilla qubit. Circuit depth remains linear in quantum technologies with finite-distance interactions between qubits. The suggested construction is related to the long-standing construction by Barenco et al. (Phys. Rev. A, 52: 3457-3467, 1995, arXiv:quant-ph/9503016), which uses a quadratic-size, quadratic-depth quantum circuit for an $n$-qubit Toffoli gate.
We provide an explicit construction of a universal gate set for continuous-variable quantum computation with microwave circuits. Such a universal set has been first proposed in quantum-optical setups, but its experimental implementation has remained elusive in that domain due to the difficulties in engineering strong nonlinearities. Here, we show that a realistic three-wave mixing microwave architecture based on the SNAIL [Frattini et al., Appl. Phys. Lett. 110, 222603 (2017)] allows us to overcome this difficulty. As an application, we show that this architecture allows for the generation of …
Construction of explicit quantum circuits follows the notion of the "standard circuit model" introduced in the solid and profound analysis of elementary gates providing quantum computation. Nevertheless the model is not always optimal (e.g. concerning the number of computational steps) and it neglects physical systems which cannot follow the "standard circuit model" analysis. We propose a computational scheme which overcomes the notion of the transposition from classical circuits providing a computation scheme with the least possible number of Hamiltonians in order to minimize the physical res…
Quantum Compiling Algorithms decompose (exactly, without approximations) an arbitrary $2^\nb$ unitary matrix acting on $\nb$ qubits, into a sequence of elementary operations (SEO). There are many possible ways of decomposing a unitary matrix into a SEO, and some of these decompositions have shorter length (are more efficient) than others. Finding an optimum (shortest) decomposition is a very hard task, and is not our intention here. A less ambitious, more doable task is to find methods for optimizing small segments of a SEO. Call these methods piecewise optimizations. Piecewise optimizations i…
We provide a detailed estimate for the logical resource requirements of the quantum linear system algorithm (QLSA) [Phys. Rev. Lett. 103, 150502 (2009)] including the recently described elaborations [Phys. Rev. Lett. 110, 250504 (2013)]. Our resource estimates are based on the standard quantum-circuit model of quantum computation; they comprise circuit width, circuit depth, the number of qubits and ancilla qubits employed, and the overall number of elementary quantum gate operations as well as more specific gate counts for each elementary fault-tolerant gate from the standard set {X, Y, Z, H, …
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 present an n-bit Toffoli gate quantum circuit based on the realization proposed by Barenco, where some of the Toffoli gates in their construction are replaced with Peres gates. This results in a significant cost reduction. Our main contribution is a quantum circuit which simulates the (m+1)-bit Toffoli gate with 32m-96 elementary quantum gates and one garbage bit which is passed unchanged. This paper is a corrected and expanded version of our recent journal publication.
Quantum error correction (QEC) is fundamental for quantum information processing but entails a substantial overhead of classically-controlled quantum operations, which can be architecturally cumbersome to accommodate. Here we discuss a novel approach to designing elementary QEC memory cells, in which all control operations are performed autonomously by an embedded optical feedback loop. Our approach is natural for nanophotonic implementations in which each qubit can be coupled to its own optical resonator, and our design for a memory cell based on the quantum bit-flip or phase-flip code requir…
The synthesis of a quantum circuit consists in decomposing a unitary matrix into a series of elementary operations. In this paper, we propose a circuit synthesis method based on the QR factorization via Householder transformations. We provide a two-step algorithm: during the first step we exploit the specific structure of a quantum operator to compute its QR factorization, then the factorized matrix is used to produce a quantum circuit. We analyze several costs (circuit size and computational time) and compare them to existing techniques from the literature. For a final quantum circuit twice a…
Formulas to calculate multivector exponentials in a base-free representation and in a orthonormal basis are presented for an arbitrary Clifford geometric algebra Cl(p,q). The formulas are based on the analysis of roots of characteristic polynomial of a multivector. Elaborate examples how to use the formulas in practice are presented. The results may be useful in the quantum circuits or in the problems of analysis of evolution of the entangled quantum states.
