We develop a circuit theory that enables us to analyze quantum measurements on a two-level system and on a continuous-variable system on an equal footing. As a measurement scheme applicable to both systems, we discuss a swapping state measurement which exchanges quantum states between the system and the measuring apparatus before the apparatus meter is read out. This swapping state measurement has an advantage in gravitational-wave detection over contractive state measurement in that the postmeasurement state of the system can be set to a prescribed one, regardless of the outcome of the measurement.
Measurement-driven quantum advantages in shallow circuits
Dahlem Center for Complex Quantum Systems, Freie Universität Berlin（柏林自由大学达勒姆复杂量子系统中心）；Helmholtz-Zentrum Berlin für Materialien und Energie（柏林亥姆霍兹材料能源中心）
Clifford circuits play an important role in quantum computation. Gottesman and Chuang proposed a gate teleportation protocol so that a quantum circuit can be implemented by the teleportation circuit with specific ancillary qubits. In particular, an $n$-qubit Clifford circuit $U$ can be implemented by preparing an ancillary stabilizer state $(I\otimes U)|Φ^+\rangle^{\otimes n}$ for teleportation and doing a Pauli correction conditioned on the measurement. In this paper, we provide an alternative procedure to implement a Clifford circuit through Pauli measurements, by preparing $O(1)$ ancillas t…
Steady-state phases in long-range measurement-only quantum circuits
Homer L. Dodge Department of Physics and Astronomy, The University of Oklahoma（霍默·L·道奇物理与天文系，俄克拉荷马大学）；Center for Quantum Research and Technology, The University of Oklahoma（量子研究与技术中心，俄克拉荷马大学）
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…
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.
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
Measurements that occur within the internal layers of a quantum circuit -- mid-circuit measurements -- are an important quantum computing primitive, most notably for quantum error correction. Mid-circuit measurements have both classical and quantum outputs, so they can be subject to error modes that do not exist for measurements that terminate quantum circuits. Here we show how to characterize mid-circuit measurements, modelled by quantum instruments, using a technique that we call quantum instrument linear gate set tomography (QILGST). We then apply this technique to characterize a dispersive…
A model of repeated quantum measurements of magnetic flux in superconducting circuits manifesting tunneling is discussed. The perturbation due to the previous measurements of magnetic flux is always present unless quantum nondemolition measurements are performed. By replacing the classical notion of noninvasivity with this condition, temporal Bell-like inequalities allows one to test the observability at the macroscopic level of the conflict between realism and quantum theory.
In this paper, we present Clifford+T gates based quantum circuit design of integer division having $n$ ancillary qubits. The proposed quantum circuit is based on restoring division algorithm. The proposed quantum circuit of integer division consists of (i) quantum circuitry of conditional addition operation, (ii) quantum circuitry of integer subtraction. To design ancillary and T-count optimized design of quantum integer division, the optimized quantum circuit design of integer conditional addition operation and integer subtraction are presented. The proposed quantum integer division circuitry…
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…
Noise in quantum systems is a major obstacle to implementing many quantum algorithms on large quantum circuits. In this work, we study the effects of noise on the Rademacher complexity of quantum circuits, which is a measure of statistical complexity that quantifies the richness of classes of functions generated by these circuits. We consider noise models that are represented by convex combinations of unitary channels and provide both upper and lower bounds for the Rademacher complexities of quantum circuits characterized by these noise models. In particular, we find a lower bound for the Rade…
We propose a classical-quantum hybrid algorithm for machine learning on near-term quantum processors, which we call quantum circuit learning. A quantum circuit driven by our framework learns a given task by tuning parameters implemented on it. The iterative optimization of the parameters allows us to circumvent the high-depth circuit. Theoretical investigation shows that a quantum circuit can approximate nonlinear functions, which is further confirmed by numerical simulations. Hybridizing a low-depth quantum circuit and a classical computer for machine learning, the proposed framework paves th…
Although many of works have been done in multivalued quantum logic synthesis, the question whether multivalued quantum circuits are more efficient than the conventional binary quantum circuits is still open. In this article we devote to the optimization of generic multivalued quantum circuits. The multivalued quantum Shannon decompositions (QSD) are improved so that the circuits obtained are asymptotically optimal for all dimensionality d. The syntheses of uniformly multifold controlled $R_y$ rotations are also optimized to make the circuits further simplified. Moreover, the theoretical lower …
Constructing general programmable circuits to be able to run any given unitary operator efficiently on a quantum processor is of fundamental importance. We present a new quantum circuit design technique resulting two general programmable circuit schemes. The circuit schemes can be used to simulate any given operator by setting the angle values in the circuit. This provides a fixed circuit design whose angles are determined from the elements of the given matrix, which can be non-unitary, in an efficient way. We also give both classical and quantum complexity analysis for these circuits and show…
Quantum algorithms and protocols are often presented as quantum circuits for a better understanding. We give a list of equivalence rules which can help in the analysis and design of quantum circuits. As example applications we study quantum teleportation and dense coding protocols in terms of a simple XOR swapping circuit and give an intuitive picture of a basic gate teleportation circuit.
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…
As the width and depth of quantum circuits implemented by state-of-the-art quantum processors rapidly increase, circuit analysis and assessment via classical simulation are becoming unfeasible. It is crucial, therefore, to develop new methods to identify significant error sources in large and complex quantum circuits. In this work, we present a technique that pinpoints the sections of a quantum circuit that affect the circuit output the most and thus helps to identify the most significant sources of error. The technique requires no classical verification of the circuit output and is thus a sca…
In this paper we present a method for minimizing reversible quantum circuits using the Quantum Operator Form (QOF); a new representation of quantum circuit and of quantum-realized reversible circuits based on the CNOT, CV and CV$^\dagger$ quantum gates. The proposed form is a quantum extension to the well known Reed-Muller but unlike the Reed-Muller form, the QOF allows the usage of different quantum gates. Therefore QOF permits minimization of quantum circuits by using properties of different gates than only the multi-control Toffoli gates. We introduce a set of minimization rules and a pseud…
This paper proposes a quantum circuit for computing the mean value from a given set of quantum states. The circuit consults a Quantum Random Access Memory to get the values of the set, and by using superposition, interference and entanglement phenomena, it can estimate the mean value in $\mathcal{O}(\frac{1}ε\log{Nd})$ complexity. The proposed quantum mean-estimator circuit has been simulated on the IBM Q Experience and the results suggest that the proposed quantum circuit can have the potential to enhance many mean-based machine learning algorithms.
A mathematical proposition with a trainable pair, operator and quantum circuit, are introduced to approximate functions expressed as cubic Taylor polynomials, numerical simulations illustrate three cases.
Quantum computation has attracted much attention, among other things, due to its potentialities to solve classical NP problems in polynomial time. For this reason, there has been a growing interest to build a quantum computer. One of the basic steps is to implement the quantum circuit able to realize a given unitary operation. This task has been solved using decomposition of unitary matrices in simpler ones till reach quantum circuits having only single-qubits and CNOTs gates. Usually the goal is to find the minimal quantum circuit able to solve a given problem. In this paper we go in a differ…
Any unitary transformation can be decomposed into a product of a group of near-trivial transformations. We investigate in details the construction of universal quantum circuit of near trivial transformations. We first construct two universal quantum circuits which can implement any single-qubit rotation $R_y(θ)$ and $R_z(θ)$ within any given precision, and then we construct universal quantum circuit implementing any single-qubit transformation within any given precision. Finally, a universal quantum circuit implementing any $n$-qubit near-trivial transformation is constructed using the univers…
Quantum circuits are time dependent diagrams describing the process of quantum computation. Usually, a quantum algorithm must be mapped into a quantum circuit. Optimal synthesis of quantum circuits is intractable and heuristic methods must be employed. With the use of heuristics, the optimality of circuits is no longer guaranteed. In this paper, we consider a local optimization technique based on templates to simplify and reduce the depth of non-optimal quantum circuits. We present and analyze templates in the general case, and provide particular details for the circuits composed of NOT, CNOT …
We find a sufficient set of equations between quantum circuits from which we can derive any other equation between stabilizer quantum circuits. To establish this result, we rely upon existing work on the completeness of the graphical ZX language for quantum processes. The complexity of the circuit equations, as opposed to the very intuitive reading of the much smaller number of ZX-equations, advocates the latter for performing computations with quantum circuits.
Despite the rapid development of quantum computing these years, state-of-the-art quantum devices still contain only a very limited number of qubits. One possible way to execute more realistic algorithms in near-term quantum devices is to employ dynamic quantum circuits, in which measurements can happen during the circuit and their outcomes are used to control other parts of the circuit. This technique can help to significantly reduce the resources required to achieve a given accuracy of a quantum algorithm. However, since this type of quantum circuits are more flexible, their verification is m…
One-way measurement based quantum computations (1WQC) may describe unitary transformations, via a composition of CPTP maps which are not all unitary themselves. This motivates the following decision problems: Is it possible to determine whether a ``quantum-to-quantum'' 1WQC procedure (having non-trivial input and output subsystems) performs a unitary transformation? Is it possible to describe precisely how such computations transform quantum states, by translation to a quantum circuit of comparable complexity? In this article, we present an efficient algorithm for transforming certain families…
We present a rigorous proof that quantum circuit algorithm can be transformed into quantum adiabatic algorithm with the exact same time complexity. This means that from a quantum circuit algorithm of $L$ gates we can construct a quantum adiabatic algorithm with time complexity of $O(L)$. Additionally, our construction shows that one may exponentially speed up some quantum adiabatic algorithms by properly choosing an evolution path.
Diagonal quantum circuits are quantum circuits comprising only diagonal gates in the computational basis. In spite of a classical feature of diagonal quantum circuits in the sense of commutativity of all gates, their computational power is highly likely to outperform classical one and they are exploited for applications in quantum informational tasks. We review computational power of diagonal quantum circuits and their applications. We focus on the computational power of instantaneous quantum polynomial-time (IQP) circuits, which are a special type of diagonal quantum circuits. We then review …
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…
This study presents a quantum circuit for estimating the pi value using arithmetic circuits and by quantum amplitude estimation. We review two types of quantum multipliers and propose quantum squaring circuits based on the multiplier as basic arithmetic circuits required for performing quantum computations. The squarer realized by a quantum adder with the gate size of $ O(n) $ requires $ O(n^2) $ gates and at least one ancillary qubits, while that realized by using quantum Fourier transform (QFT) requires $ O(n^3) $ gates without ancillary qubit. The proposed quantum circuit to estimate pi is …
We construct quantum circuits for solving one-dimensional Schrödinger equations. Simulations of three typical examples, i.e., harmonic oscillator, square-well and Coulomb potential, show that reasonable results can be obtained with eight qubits. Our simulations show that simple quantum circuits can solve the standard quantum mechanical problems.
Review of the probability representation of quantum mechanics and the symplectic tomography approach are presented. The examples of Gaussian states of nanoelectric circuit, Josephson junction, and two interacting high-quality resonant circuits are considered. The Shannon entropy, quantum information, fidelity, and purity of quantum states in the tomographic representation of quantum mechanics are studied.
There are quantum circuit identities that simplify quantum circuits, reducing the effort needed physically to implement them. This paper constructs all identities made from 3 or fewer operations taken from a common set of one qubit operations, and explains how they may be used to simplify the cost of constructing quantum circuit identities.
In this paper we address the problem of translating one-way quantum computation (1WQC) into the circuit model. We start by giving a straightforward circuit representation of any 1WQC, at the cost of introducing many ancilla wires. We then propose a set of four simple circuit identities that explore the relationship between the entanglement resource and correction structure of a 1WQC, allowing one to obtain equivalent circuits acting on fewer qubits. We conclude with some examples and a discussion of open problems.
Circuit design for quantum machine learning remains a formidable challenge. Inspired by the applications of tensor networks across different fields and their novel presence in the classical machine learning context, one proposed method to design variational circuits is to base the circuit architecture on tensor networks. Here, we comprehensively describe tensor-network quantum circuits and how to implement them in simulations. This includes leveraging circuit cutting, a technique used to evaluate circuits with more qubits than those available on current quantum devices. We then illustrate the …
Translations between the quantum circuit model and the measurement-based one-way model are useful for verification and optimisation of quantum computations. They make crucial use of a property known as gflow. While gflow is defined for one-way computations allowing measurements in three different planes of the Bloch sphere, most research so far has focused on computations containing only measurements in the XY-plane. Here, we give the first circuit-extraction algorithm to work for one-way computations containing measurements in all three planes and having gflow. The algorithm is efficient and …
The quantum circuit layout (QCL) problem is to map a quantum circuit such that the constraints of the device are satisfied. We introduce a quantum circuit mapping heuristic, QXX, and its machine learning version, QXX-MLP. The latter infers automatically the optimal QXX parameter values such that the layed out circuit has a reduced depth. In order to speed up circuit compilation, before laying the circuits out, we are using a Gaussian function to estimate the depth of the compiled circuits. This Gaussian also informs the compiler about the circuit region that influences most the resulting circu…
Noise mitigation and reduction will be crucial for obtaining useful answers from near-term quantum computers. In this work, we present a general framework based on machine learning for reducing the impact of quantum hardware noise on quantum circuits. Our method, called noise-aware circuit learning (NACL), applies to circuits designed to compute a unitary transformation, prepare a set of quantum states, or estimate an observable of a many-qubit state. Given a task and a device model that captures information about the noise and connectivity of qubits in a device, NACL outputs an optimized circ…
Multipartite entanglement is of important resources for quantum communication and quantum computation. Our goal in this paper is to characterize general multipartite entangled states according to shallow quantum circuits. We firstly prove any genuinely multipartite entanglement on finite-dimensional spaces can be generated by using 2-layer shallow quantum circuit consisting of two biseparable quantum channels, which the smallest nontrivial circuit depth in the shallow quantum circuit model. We further propose a semi-device-independent entanglement model depending on the local connection abilit…
