We exploit a recently constructed mapping between quantum circuits and graphs in order to prove that circuits corresponding to certain planar graphs can be efficiently simulated classically. The proof uses an expression for the Ising model partition function in terms of quadratically signed weight enumerators (QWGTs), which are polynomials that arise naturally in an expansion of quantum circuits in terms of rotations involving Pauli matrices. We combine this expression with a known efficient classical algorithm for the Ising partition function of any planar graph in the absence of an external magnetic field, and the Robertson-Seymour theorem from graph theory. We give as an example a set of quantum circuits with a small number of non-nearest neighbor gates which admit an efficient classical simulation.
We present a simple construction that maps quantum circuits to graphs and vice-versa. Inspired by the results of D.A. Lidar linking the Ising partition function with quadratically signed weight enumerators (QWGTs), we also present a BQP-complete problem for the additive approximation of a function over hypergraphs related to the generating function of Eulerian subgraphs for ordinary graphs. We discuss connections with the Ising partition function.
We present an exact simulation of a one-dimensional transverse Ising spin chain with a quantum computer. We construct an efficient quantum circuit that diagonalizes the Ising Hamiltonian and allows to obtain all eigenstates of the model by just preparing the computational basis states. With an explicit example of that circuit for $n=4$ spins, we compute the expected value of the ground state magnetization, the time evolution simulation and provide a method to also simulate thermal evolution. All circuits are run in IBM and Rigetti quantum devices to test and compare them qualitatively.
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.
It is imperative to compile quantum circuits for Noisy Intermediate-Scale Quantum (NISQ) devices because of the limited connectivity of physical qubits and the high error rates of gate operations. One of the most critical steps in quantum circuit compilation is qubit routing, an NP-Hard problem that involves placing and moving logical qubits to minimize compilation overhead. In this study, we propose ISing mAchine Assisted Quantum compiler (ISAAQ) to perform qubit routing with Ising machines, which can efficiently solve Quadratic Unconstrained Binary Optimization (QUBO) problems. ISAAQ accurat…
We propose a network of open-dissipative quantum oscillators with optical error correction circuits. In the proposed network, the squeezed/anti-squeezed vacuum states of the constituent optical parametric oscillators below the threshold establish quantum correlations through optical mutual coupling, while collective symmetry breaking is induced above the threshold as a decision-making process. This initial search process is followed by a chaotic solution search step facilitated by the optical error correction feedback. As an optical hardware technology, the proposed coherent Ising machine (CIM…
We show how a quantum Ising spin chain in a time-dependent transverse magnetic field can be simulated and experimentally probed in the framework of circuit QED with current technology. The proposed setup provides a new platform for observing the nonequilibrium dynamics of interacting many-body systems. We calculate its spectrum to offer a guideline for its initial experimental characterization. We demonstrate that quench dynamics and the propagation of localized excitations can be observed with the proposed setup and discuss further possible applications and modifications of this circuit QED q…
We study several dynamical properties of a recently proposed implementation of the quantum transverse-field Ising chain in the framework of circuit QED. Particular emphasis is placed on the effects of disorder on the nonequilibrium behavior of the system. We show that small amounts of fabrication-induced disorder in the system parameters do not jeopardize the observation of previously-predicted phenomena. Based on a numerical extraction of the mean free path of the system, we also provide a simple quantitative estimate for certain disorder effects on the nonequilibrium dynamics of the circuit …
We explore ideas for generating sounds and eventually music by using quantum devices in the NISQ era using quantum circuits. In particular, we first consider a concept for a "qeyboard", i.e. a quantum keyboard, where the real-time behaviour of expectation values using a time evolving quantum circuit can be associated to sound features like intensity, frequency and tone. Then, we examine how these properties can be extracted from physical quantum systems, taking the Ising model as an example. This can be realized by measuring physical quantities of the quantum states of the system, e.g. the ene…
Optimally mapping a parallel application to compute and communication resources is increasingly important as both system size and heterogeneity increase. A similar mapping problem exists in gate-based quantum computing where the objective is to map tasks to gates in a topology-aware fashion. This is an NP-complete graph isomorphism problem, and existing task assignment approaches are either heuristic or based on physical optimization algorithms, providing different speed and solution quality trade-offs. Ising machines such as quantum and digital annealers have recently become available and off…
We describe an efficient approximation algorithm for evaluating the ground-state energy of the classical Ising Hamiltonian with linear terms on an arbitrary planar graph. The running time of the algorithm grows linearly with the number of spins and exponentially with 1/epsilon, where epsilon is the worst-case relative error. This result contrasts the well known fact that exact computation of the ground-state energy for the two-dimensional Ising spin glass model is NP-hard. We also present a classical approximation algorithm for the Local Hamiltonian Problem or Quantum Ising Spin Glass problem …
A scheme for measuring complex temperature partition functions of Ising models is introduced. In the context of ordered qubit registers this scheme finds a natural translation in terms of global operations, and single particle measurements on the edge of the array. Two applications of this scheme are presented. First, through appropriate Wick rotations, those amplitudes can be analytically continued to yield estimates for partition functions of Ising models. Bounds on the estimation error, valid with high confidence, are provided through a central-limit theorem, which validity extends beyond t…
Instantaneous quantum polynomial-time (IQP) computation is a class of quantum computation consisting only of commuting two-qubit gates and is not universal in the sense of standard quantum computation. Nevertheless, it has been shown that if there is a classical algorithm that can simulate IQP efficiently, the polynomial hierarchy (PH) collapses at the third level, which is highly implausible. However, the origin of the classical intractability is still less understood. Here we establish a relationship between IQP and computational complexity of the partition functions of Ising models. We appl…
The quantum theory of coherent Ising machines, based on degenerate optical parametric oscillators and measurement-feedback circuits, is developed using the positive $P(α,β)$ representation of the density operator and the master equation. The theory is composed of the c-number stochastic differential equations for describing open dissipative quantum dynamics and the replicator dynamics equations for handling measurement-induced collapse of the density operator. We apply the present theory to simulate two simple Ising spin models and elucidate the unique features of this computing machine.
The notion of compressed quantum computation is employed to simulate the Ising interaction of a 1D--chain consisting out of $n$ qubits using the universal IBM cloud quantum computer running on $\log(n)$ qubits. The external field parameter that controls the quantum phase transition of this model translates into particular settings of the quantum gates that generate the circuit. We measure the magnetization, which displays the quantum phase transition, on a two--qubit system, which simulates a four--qubit Ising chain, and show its agreement with the theoretical prediction within a certain error…
In [R. Jozsa, B. Kraus, A. Miyake, J. Watrous, Proc. R. Soc. A {\bf 466}, 809-830 (2010)] it has been shown that a match gate circuit running on n qubits can be compressed to a universal quantum computation on \log(n)+3 qubits. Here, we show how this compression can be employed to simulate the Ising interaction of a 1D--chain consisting out of n qubits using a universal quantum computer running on log(n) qubits. We demonstrate how the adiabatic evolution can be realized on this exponentially smaller system and how the magnetization, which shows a quantum phase transition, can be measured.
We propose an efficient algorithm for partitioning Pauli strings into subgroups, which can be simultaneously measured in a single quantum circuit. Our partitioning algorithm drastically reduces the total number of measurements in a variational quantum eigensolver for a quantum chemistry, one of the most promising applications of quantum computing. The algorithm is based on the Ising model optimization problem, which can be quickly solved using an Ising machine. We develop an algorithm that is applicable to problems with sizes larger than the maximum number of variables that an Ising machine ca…
We propose the implementation of a digital quantum simulation of spin chains coupled to bosonic field modes in superconducting circuits. Gates with high fidelities allows one to simulate a variety of Ising magnetic pairing interactions with transverse field, Tavis-Cummings interaction between spins and a bosonic mode, and a spin model with three-body terms. We analyze the feasibility of the implementation in realistic circuit quantum electrodynamics setups, where the interactions are either realized via capacitive couplings or mediated by microwave resonators.
The ferromagnetic state of an Ising chain can represent a two-fold degenerate subspace or equivalently a logical qubit which is protected from excitations by an energy gap. We study a a braiding-like exchange operation through the movement of the state in the qubit subspace which resembles that of the localized edge modes in a Kitaev chain. The system consists of two Ising chains in a 1D geometry where the operation is simulated through the adiabatic time evolution of the ground state. The time evolution is implemented via the Suzuki-Trotter expansion on basic single- and two-qubit quantum gat…
In a typical quantum annealing protocol, the system starts with a transverse field Hamiltonian which is gradually turned off and replaced by a longitudinal Ising Hamiltonian. The ground state of the Ising Hamiltonian encodes the solution to the computational problem of interest, and the state overlap with this ground state gives the success probability of the annealing protocol. The form of the annealing schedule can have a significant impact on the ground state overlap at the end of the anneal, so precise control over these annealing schedules can be a powerful tool for increasing success pro…
University of Edinburgh（爱丁堡大学）；CNRS（法国国家科学研究中心）；École Normale Supérieure（巴黎高等师范学院）；Sorbonne Université（索邦大学）
The search for an application of near-term quantum devices is widespread. Quantum Machine Learning is touted as a potential utilisation of such devices, particularly those which are out of the reach of the simulation capabilities of classical computers. In this work, we propose a generative Quantum Machine Learning Model, called the Ising Born Machine (IBM), which we show cannot, in the worst case, and up to suitable notions of error, be simulated efficiently by a classical device. We also show this holds for all the circuit families encountered during training. In particular, we explore quant…
Current quantum annealing (QA) hardware suffers from practical limitations such as finite temperature, sparse connectivity, small qubit numbers, and control error. We propose new algorithms for mapping boolean constraint satisfaction problems (CSPs) onto QA hardware mitigating these limitations. In particular we develop a new embedding algorithm for mapping a CSP onto a hardware Ising model with a fixed sparse set of interactions, and propose two new decomposition algorithms for solving problems too large to map directly into hardware. The mapping technique is locally-structured, as hardware c…
In this article we present a full description of the quantum Kerr Ising model---a linear optical network of parametrically pumped Kerr non-linearities. We consider the non-dissapative Kerr Ising model and, using variational techniques, show that the energy spectrum is primarily determined by the adjacency matrix in the Ising model and exhibits highly non-classical cat like eigenstates. We then introduce dissipation to give a quantum mechanical treatment of the measurement process based on homodyne detection via the conditional stochastic Schrodinger equation. Finally we identify a quantum adva…
We study the complexity of classically sampling from the output distribution of an Ising spin model, which can be implemented naturally in a variety of atomic, molecular, and optical systems. In particular, we construct a specific example of an Ising Hamiltonian that, after time evolution starting from a trivial initial state, produces a particular output configuration with probability very nearly proportional to the square of the permanent of a matrix with arbitrary integer entries. In a similar spirit to BosonSampling, the ability to sample classically from the probability distribution induc…
In breakthrough work, Bravyi, Gosset, and König (BGK) [Science, 2018] unconditionally proved that constant depth quantum circuits are more powerful than their classical counterparts. Their result is equivalent to saying that a particular family of constant depth quantum circuits takes classical circuits at least $Ω(\log n)$ depth to "simulate", in a certain sense. In our paper, we formalise their sense of simulation, which we call "possibilistic simulation" or "p-simulation", and construct explicit classical circuits that can p-simulate any depth-$d$ quantum circuit with Clifford and $t$ $T$-g…
The Quantum Approximate Optimization Algorithm (QAOA) is a standard method for combinatorial optimization with a gate-based quantum computer. The QAOA consists of a particular ansatz for the quantum circuit architecture, together with a prescription for choosing the variational parameters of the circuit. We propose modifications to both. First, we define the Gibbs objective function and show that it is superior to the energy expectation value for use as an objective function in tuning the variational parameters. Second, we describe an Ansatz Architecture Search (AAS) algorithm for searching th…
We show how quantum metrology protocols that seek to estimate the parameters of a Hamiltonian that exhibits a quantum phase transition can be efficiently simulated on an exponentially smaller quantum computer. Specifically, by exploiting the fact that the ground state of such a Hamiltonian changes drastically around its phase transition point, we construct a suitable observable from which one can estimate the relevant parameters of the Hamiltonian with Heisenberg scaling precision. We then show how, for the one-dimensional Ising Hamiltonian with transverse magnetic field acting on N spins, suc…
We propose a near-term quantum simulator based on the fluxonium qubits inductively coupled to form a chain. This system provides long coherence time, large anharmonicity, and strong coupling, making it suitable to study Ising spin models. At the half-flux quantum sweet spot, the system is described by the transverse field Ising model (TFIM). We evaluate the propagation of qubit excitations through the system. As disorder increases, the excitations become localized. A single qubit measurement using the circuit QED methods is sufficient to identify localization transition without introducing tun…
The study of quantum circuits composed of commuting gates is particularly useful to understand the delicate boundary between quantum and classical computation. Indeed, while being a restricted class, commuting circuits exhibit genuine quantum effects such as entanglement. In this paper we show that the computational power of commuting circuits exhibits a surprisingly rich structure. First we show that every 2-local commuting circuit acting on d-level systems and followed by single-qudit measurements can be efficiently simulated classically with high accuracy. In contrast, we prove that such st…
Two objects can be distinguished if they have different measurable properties. Thus, distinguishability depends on the Physics of the objects. In considering graphs, we revisit the Ising model as a framework to define physically meaningful spectral invariants. In this context, we introduce a family of refinements of the classical spectrum and consider the quantum partition function. We demonstrate that the energy spectrum of the quantum Ising Hamiltonian is a stronger invariant than the classical one without refinements. For the purpose of implementing the related physical systems, we perform …
Quantum computation is a continuously growing research area which is based on nature and resources of quantum mechanics, as superposition and entanglement. In its quantum circuits version, the use of convenient and appropriate gates is essential. But while those gates adopt convenient forms for computational algorithms, their design depends on specific quantum systems and stu? being used. These gates need manage quantum systems based on physical interactions ruled by quantum Hamiltonians. With this, gates design is restricted to properties and limitations of interactions and physical elements …
This note introduces "Q-marginals", which are quantum states encoding some probability distribution in a manner suitable for use in Quantum Monte Carlo Integration (QMCI), and shows that these can be prepared directly from a classical circuit sampling for the probability distribution of interest. This result is important as the quantum advantage in Monte Carlo integration is in the form of a reduction in the number of uses of a quantum state encoding the probability distribution (in QMCI) relative to the number of samples that would be required in classical MCI -- hence it only translates into…
We consider Gaussian quantum circuits supplemented with non-Gaussian input states and derive sufficient conditions for efficient classical strong simulation of these circuits. In particular, we generalise the stellar representation of continuous-variable quantum states to the multimode setting and relate the stellar rank of the input non-Gaussian states, a recently introduced measure of non-Gaussianity, to the cost of evaluating classically the output probability densities of these circuits. Our results have consequences for the strong simulability of a large class of near-term continuous-vari…
Recently, constant-depth quantum circuits are proved more powerful than their classical counterparts at solving certain problems, e.g., the two-dimensional (2D) hidden linear function (HLF) problem regarding a symmetric binary matrix. To further investigate the boundary between classical and quantum computing models, in this work we propose a high-performance two-stage classical scheme to solve a full-sampling variant of the 2D HLF problem, which combines traditional classical parallel algorithms and a gate-based classical circuit model together for exactly simulating the target shallow quantu…
Quantum circuits which perform integer arithmetic could potentially outperform their classical counterparts. In this paper, a quantum circuit is considered which performs a specific computational pattern on classically represented integers to accelerate the computation. Such a hybrid circuit could be embedded in a conventional computer architecture as a quantum device or accelerator. In particular, a quantum multiply-add circuit (QMAC) using a Quantum Fourier Transform (QFT) is proposed which can perform the calculation on conventional integers faster than its conventional counterpart. Whereas…
High-connectivity circuits are a major roadblock for current quantum hardware. We propose a hybrid classical-quantum algorithm to simulate such circuits without swap-gate ladders. As main technical tool, we introduce quantum-classical-quantum interfaces. These replace an experimentally problematic gate (e.g. a long-range one) by single-qubit random measurements followed by state-preparations sampled according to a classical quasi-probability simulation of the noiseless gate. Each interface introduces a multiplicative statistical overhead which is remarkably independent of the on-chip qubit dis…
We propose a realizable circuit QED architecture for engineering states of a superconducting resonator off-resonantly coupled to an ancillary superconducting qubit. The qubit-resonator dispersive interaction together with a microwave drive applied to the qubit gives rise to a Kerr resonator with two-photon driving that enables us to efficiently engineer the quantum state of the resonator such as generation of the Schrodinger cat states for resonator-based universal quantum computation. Moreover, the presented architecture is easily scalable for solving optimization problem mapped into the Isin…
We consider the long-time limit of out-of-time-order correlators (OTOCs) in two classes of quantum lattice models with time evolution governed by local unitary quantum circuits and maximal butterfly velocity $v_{B} = 1$. Using a transfer matrix approach, we present analytic results for the long-time value of the OTOC on and inside the light cone. First, we consider `dual-unitary' circuits with various levels of ergodicity, including the integrable and non-integrable kicked Ising model, where we show exponential decay away from the light cone and relate both the decay rate and the long-time val…
In this paper we study the ways to use a global entangling operator to efficiently implement circuitry common to a selection of important quantum algorithms. In particular, we focus on the circuits composed with global Ising entangling gates and arbitrary addressable single-qubit gates. We show that under certain circumstances the use of global operations can substantially improve the entangling gate count.
The coherent Ising machine is expected to find a near-optimal solution in various combinatorial optimization problems, which has been experimentally confirmed with optical parametric oscillators (OPOs) and a field programmable gate array (FPGA) circuit. The similar mathematical models were proposed three decades ago by J. J. Hopfield, et al. in the context of classical neural networks. In this article, we compare the computational performance of both models.
We prove that the 2D Ising model is complete in the sense that the partition function of any classical q-state spin model (on an arbitrary graph) can be expressed as a special instance of the partition function of a 2D Ising model with complex inhomogeneous couplings and external fields. In the case where the original model is an Ising or Potts-type model, we find that the corresponding 2D square lattice requires only polynomially more spins w.r.t the original one, and we give a constructive method to map such models to the 2D Ising model. For more general models the overhead in system size ma…
We present a polynomial time approximation scheme (PTAS) for the minimum value of the classical Ising Hamiltonian with linear terms on the Chimera graph structure as defined in the recent work of McGeoch and Wang. The result follows from a direct application of the techniques used by Bansal, Bravyi and Terhal who gave a PTAS for the same problem on planar and, in particular, grid graphs. We also show that on Chimera graphs, the trivial lower bound is within a constant factor of the optimum.
We present a quantitative semi-classical theory for the non-equilibrium dynamics of transverse Ising chains after quantum quenches, in particular sudden changes of the transverse field strength. We obtain accurate predictions for the quench dependent relaxation times and correlation lengths, and also about the recurrence times and quasiperiodicity of time dependent correlations in finite systems with open or periodic boundary conditions. We compare the quantiative predictions of our semi-classical theory (local magnetization, equal time bulk-bulk and surface-to-bulk correlations, and bulk auto…
