Solving Systems of Linear Equations with a Superconducting Quantum Processor

Addressing quantum's "fine print" with efficient state preparation and information extraction for quantum algorithms and geologic fracture networks

Quantum algorithms provide an exponential speedup for solving certain classes of linear systems, including those that model geologic fracture flow. However, this revolutionary gain in efficiency does not come without difficulty. Quantum algorithms require that problems satisfy not only algorithm-specific constraints, but also application-specific ones. Otherwise, the quantum advantage carefully attained through algorithmic ingenuity can be entirely negated. Previous work addressing quantum algorithms for geologic fracture flow has illustrated core algorithmic approaches while incrementally removing assumptions. This work addresses two further requirements for solving geologic fracture flow systems with quantum algorithms: efficient system state preparation and efficient information extraction. Our approach to addressing each is consistent with an overall exponential speed-up.

A Variational Quantum Linear Solver Application to Discrete Finite-Element Methods

Finite-element methods are industry standards for finding numerical solutions to partial differential equations. However, the application scale remains pivotal to the practical use of these methods, even for modern-day supercomputers. Large, multi-scale applications, for example, can be limited by their requirement of prohibitively large linear system solutions. It is therefore worthwhile to investigate whether near-term quantum algorithms have the potential for offering any kind of advantage over classical linear solvers. In this study, we investigate the recently proposed variational quantum linear solver (VQLS) for discrete solutions to partial differential equations. This method was found to scale polylogarithmically with the linear system size, and the method can be implemented using shallow quantum circuits on noisy intermediate-scale quantum (NISQ) computers. Herein, we utilize the hybrid VQLS to solve both the steady Poisson equation and the time-dependent heat and wave equations.

Quantum algorithms for geologic fracture networks

Solving large systems of equations is a challenge for modeling natural phenomena, such as simulating subsurface flow. To avoid systems that are intractable on current computers, it is often necessary to neglect information at small scales, an approach known as coarse-graining. For many practical applications, such as flow in porous, homogenous materials, coarse-graining offers a sufficiently-accurate approximation of the solution. Unfortunately, fractured systems cannot be accurately coarse-grained, as critical network topology exists at the smallest scales, including topology that can push the network across a percolation threshold. Therefore, new techniques are necessary to accurately model important fracture systems. Quantum algorithms for solving linear systems offer a theoretically-exponential improvement over their classical counterparts, and in this work we introduce two quantum algorithms for fractured flow. The first algorithm, designed for future quantum computers which operate without error, has enormous potential, but we demonstrate that current hardware is too noisy for adequate performance. The second algorithm, designed to be noise resilient, already performs well for problems of small to medium size (order 10-1000 nodes), which we demonstrate experimentally and explain theoretically. We expect further improvements by leveraging quantum error mitigation and preconditioning.

Solving Complex-Valued Time-Varying Linear Matrix Equations via QR Decomposition With Applications to Robotic Motion Tracking and on Angle-of-Arrival Localization

The problem of solving linear equations is considered as one of the fundamental problems commonly encountered in science and engineering. In this article, the complex-valued time-varying linear matrix equation (CVTV-LME) problem is investigated. Then, by employing a complex-valued, time-varying QR (CVTVQR) decomposition, the zeroing neural network (ZNN) method, equivalent transformations, Kronecker product, and vectorization techniques, we propose and study a CVTVQR decomposition-based linear matrix equation (CVTVQR-LME) model. In addition to the usage of the QR decomposition, the further advantage of the CVTVQR-LME model is reflected in the fact that it can handle a linear system with square or rectangular coefficient matrix in both the matrix and vector cases. Its efficacy in solving the CVTV-LME problems have been tested in a variety of numerical simulations as well as in two applications, one in robotic motion tracking and the other in angle-of-arrival localization.

Quantum machine learning for chemistry and physics

Machine learning (ML) has emerged as a formidable force for identifying hidden but pertinent patterns within a given data set with the objective of subsequent generation of automated predictive behavior. In recent years, it is safe to conclude that ML and its close cousin, deep learning (DL), have ushered in unprecedented developments in all areas of physical sciences, especially chemistry. Not only classical variants of ML, even those trainable on near-term quantum hardwares have been developed with promising outcomes. Such algorithms have revolutionized materials design and performance of photovoltaics, electronic structure calculations of ground and excited states of correlated matter, computation of force-fields and potential energy surfaces informing chemical reaction dynamics, reactivity inspired rational strategies of drug designing and even classification of phases of matter with accurate identification of emergent criticality. In this review we shall explicate a subset of such topics and delineate the contributions made by both classical and quantum computing enhanced machine learning algorithms over the past few years. We shall not only present a brief overview of the well-known techniques but also highlight their learning strategies using statistical physical insight. The objective of the review is not only to foster exposition of the aforesaid techniques but also to empower and promote cross-pollination among future research in all areas of chemistry which can benefit from ML and in turn can potentially accelerate the growth of such algorithms.

Synthesizing Five-Body Interaction in a Superconducting Quantum Circuit

Synthesizing many-body interaction Hamiltonians is a central task in quantum simulation. However, it is challenging to synthesize Hamiltonians that have more than two spins in a single term. Here we synthesize m-body spin-exchange Hamiltonians with m up to 5 in a superconducting quantum circuit by simultaneously exciting multiple independent qubits with time-energy correlated photons generated from a qudit. The dynamic evolution of the m-body interaction is governed by the Rabi oscillation between two m-spin states, in which the states of each spin are different. We demonstrate the scalability of our approach by comparing the influence of noises on the three-, four- and five-body interaction and building a many-body Mach-Zehnder interferometer which potentially has a Heisenberg-limit sensitivity. This study paves a way for quantum simulation involving many-body interaction Hamiltonians such as lattice gauge theories in quantum circuits.

Experimental Quantum Principal Component Analysis via Parametrized Quantum Circuits

Principal component analysis (PCA) is a widely applied but rather time-consuming tool in machine learning techniques. In 2014, Lloyd, Mohseni, and Rebentrost proposed a quantum PCA (qPCA) algorithm [Lloyd, Mohseni, and Rebentrost, Nat. Phys. 10, 631 (2014)NPAHAX1745-247310.1038/nphys3029] that still lacks experimental demonstration due to the experimental challenges in preparing multiple quantum state copies and implementing quantum phase estimations. Here, we propose a new qPCA algorithm using the hybrid classical-quantum control, where parameterized quantum circuits are optimized with simple measurement observables, which significantly reduces the experimental complexity. As one important PCA application, we implement a human face recognition process using the images from the Yale Face Dataset. By training our quantum processor, the eigenface information in the training dataset is encoded into the parameterized quantum circuit, and the quantum processor learns to recognize new face images from the test dataset with high fidelities. Our work paves a new avenue toward the study of qPCA applications in theory and experiment.

Fusing the single-excitation subspace with [Formula: see text]

There is a tremendous interest in developing practical applications for noisy intermediate-scale quantum processors without the overhead required by full error correction. Near-term quantum information processing is especially challenging within the standard gate model, as algorithms quickly lose fidelity as the problem size and circuit depth grow. This has lead to a number of non-gate-model approaches such as analog quantum simulation and quantum annealing. These come with specific hardware requirements that are different than that of a universal gate-based quantum computer. We have previously proposed an approach called the single-excitation subspace (SES) method, which uses a complete graph of superconducting qubits with tunable coupling. Without error correction the SES method is not scalable, but it offers several algorithmic components with constant depth, which is highly desirable for near-term use. The challenge of the SES method is that it requires a physical qubit for every basis state in the computer's Hilbert space. This imposes exponentially large resource costs for algorithms using registers of ancillary qubits, as each ancilla would double the required graph size. Here we show how to circumvent this doubling by leaving the SES and fusing it with a multi-ancilla Hilbert space. Specifically, we implement the tensor product of an SES register holding "data" with one or more ancilla qubits, which are able to independently control arbitrary [Formula: see text] unitary operations on the data in a constant number of steps. This enables a hybrid form of quantum computation where fast SES operations are performed on the data, traditional logic gates and measurements are performed on the ancillas, and controlled-unitaries act between. As example applications, we give ancilla-assisted SES implementations of quantum phase estimation and the quantum linear system solver of Harrow, Hassidim, and Lloyd.

Simultaneous Excitation of Two Noninteracting Atoms with Time-Frequency Correlated Photon Pairs in a Superconducting Circuit

We report the first observation of simultaneous excitation of two noninteracting atoms by a pair of time-frequency correlated photons in a superconducting circuit. The strong coupling regime of this process enables the synthesis of a three-body interaction Hamiltonian, which allows the generation of the tripartite Greenberger-Horne-Zeilinger state in a single step with a fidelity as high as 0.95. We further demonstrate the inhibition of the simultaneous two-atom excitation by continuously measuring whether the first photon is emitted. This work provides a new route in synthesizing many-body interaction Hamiltonian and coherent control of entanglement.

Generation of multicomponent atomic Schrödinger cat states of up to 20 qubits

Multipartite entangled states are crucial for numerous applications in quantum information science. However, the generation and verification of multipartite entanglement on fully controllable and scalable quantum platforms remains an outstanding challenge. We report the deterministic generation of an 18-qubit Greenberger-Horne-Zeilinger (GHZ) state and multicomponent atomic Schrödinger cat states of up to 20 qubits on a quantum processor, which features 20 superconducting qubits, also referred to as artificial atoms, interconnected by a bus resonator. By engineering a one-axis twisting Hamiltonian, the system of qubits, once initialized, coherently evolves to multicomponent atomic Schrödinger cat states-that is, superpositions of atomic coherent states including the GHZ state-at specific time intervals as expected. Our approach on a solid-state platform should not only stimulate interest in exploring the fundamental physics of quantum many-body systems, but also enable the development of applications in practical quantum metrology and quantum information processing.

Hybrid classical-quantum linear solver using Noisy Intermediate-Scale Quantum machines

We propose a realistic hybrid classical-quantum linear solver to solve systems of linear equations of a specific type, and demonstrate its feasibility with Qiskit on IBM Q systems. This algorithm makes use of quantum random walk that runs in [Formula: see text](N log(N)) time on a quantum circuit made of [Formula: see text](log(N)) qubits. The input and output are classical data, and so can be easily accessed. It is robust against noise, and ready for implementation in applications such as machine learning.

Propagation and Localization of Collective Excitations on a 24-Qubit Superconducting Processor

Superconducting circuits have emerged as a powerful platform of quantum simulation, especially for emulating the dynamics of quantum many-body systems, because of their tunable interaction, long coherence time, and high-precision control. Here in experiments, we construct a Bose-Hubbard ladder with a ladder array of 20 qubits on a 24-qubit superconducting processor. We investigate theoretically and demonstrate experimentally the dynamics of single- and double-excitation states with distinct behaviors, indicating the uniqueness of the Bose-Hubbard ladder. We observe the linear propagation of photons in the single-excitation case, satisfying the Lieb-Robinson bounds. The double-excitation state, initially placed at the edge, localizes; while placed in the bulk, it splits into two single-excitation modes spreading linearly toward two boundaries, respectively. Remarkably, these phenomena, studied both theoretically and numerically as unique properties of the Bose-Hubbard ladder, are represented coherently by pairs of controllable qubits in experiments. Our results show that collective excitations, as a single mode, are not free. This work paves the way to simulation of exotic logic particles by subtly encoding physical qubits and exploration of rich physics by superconducting circuits.

Quantum annealing for systems of polynomial equations

Numerous scientific and engineering applications require numerically solving systems of equations. Classically solving a general set of polynomial equations requires iterative solvers, while linear equations may be solved either by direct matrix inversion or iteratively with judicious preconditioning. However, the convergence of iterative algorithms is highly variable and depends, in part, on the condition number. We present a direct method for solving general systems of polynomial equations based on quantum annealing, and we validate this method using a system of second-order polynomial equations solved on a commercially available quantum annealer. We then demonstrate applications for linear regression, and discuss in more detail the scaling behavior for general systems of linear equations with respect to problem size, condition number, and search precision. Finally, we define an iterative annealing process and demonstrate its efficacy in solving a linear system to a tolerance of 10-8.

Genuine 12-Qubit Entanglement on a Superconducting Quantum Processor

We report the preparation and verification of a genuine 12-qubit entanglement in a superconducting processor. The processor that we designed and fabricated has qubits lying on a 1D chain with relaxation times ranging from 29.6 to 54.6  μs. The fidelity of the 12-qubit entanglement was measured to be above 0.5544±0.0025, exceeding the genuine multipartite entanglement threshold by 21 statistical standard deviations. After thermal cycling, the 12-qubit state fidelity was further improved to be above 0.707±0.008. Our entangling circuit to generate linear cluster states is depth invariant in the number of qubits and uses single- and double-qubit gates instead of collective interactions. Our results are a substantial step towards large-scale random circuit sampling and scalable measurement-based quantum computing.

Hybrid quantum linear equation algorithm and its experimental test on IBM Quantum Experience

We propose a hybrid quantum algorithm based on the Harrow-Hassidim-Lloyd (HHL) algorithm for solving a system of linear equations. In this paper, we show that our hybrid algorithm can reduce a circuit depth from the original HHL algorithm by means of a classical information feed-forward after the quantum phase estimation algorithm, and the results of the hybrid algorithm are identical to those of the HHL algorithm. In addition, it is experimentally examined with four qubits in the IBM Quantum Experience setups, and the experimental results of our algorithm show higher accurate performance on specific systems of linear equations than that of the HHL algorithm.

Solving matrix equations in one step with cross-point resistive arrays

Conventional digital computers can execute advanced operations by a sequence of elementary Boolean functions of 2 or more bits. As a result, complicated tasks such as solving a linear system or solving a differential equation require a large number of computing steps and an extensive use of memory units to store individual bits. To accelerate the execution of such advanced tasks, in-memory computing with resistive memories provides a promising avenue, thanks to analog data storage and physical computation in the memory. Here, we show that a cross-point array of resistive memory devices can directly solve a system of linear equations, or find the matrix eigenvectors. These operations are completed in just one single step, thanks to the physical computing with Ohm's and Kirchhoff's laws, and thanks to the negative feedback connection in the cross-point circuit. Algebraic problems are demonstrated in hardware and applied to classical computing tasks, such as ranking webpages and solving the Schrödinger equation in one step.

Simplified experimental scheme of quantum algorithm for solving linear equations with single photons

Linear equations play an important role in nearly all fields of research in science and engineering. Recent studies have shown that quantum algorithms for solving linear systems of algebraic equations provide exponential speedups over their classical counterparts. Here, we present a simplified experimental scheme for implementing the quantum algorithm for solving linear equations with multiple degrees of freedom (DoF) of single photon. Compared with the previous studies, our scheme not only presents excellent efficiency, it requires fewer photons to solve the same problem. Our work highlights the potential applications of multiple DoFs of single photon in quantum information processes.

Dephasing-Insensitive Quantum Information Storage and Processing with Superconducting Qubits

A central task towards building a practical quantum computer is to protect individual qubits from decoherence while retaining the ability to perform high-fidelity entangling gates involving arbitrary two qubits. Here we propose and demonstrate a dephasing-insensitive procedure for storing and processing quantum information in an all-to-all connected superconducting circuit involving multiple frequency-tunable qubits, each of which can be controllably coupled to any other through a central bus resonator. Although it is generally believed that the extra frequency tunability enhances the control freedom but induces more dephasing impact for superconducting qubits, our results show that any individual qubit can be dynamically decoupled from dephasing noise by applying a weak continuous and resonant driving field whose phase is reversed in the middle of the pulse. More importantly, we demonstrate a new method for realizing a two-qubit phase gate with inherent dynamical decoupling via the combination of continuous driving and qubit-qubit swapping coupling. We find that the weak continuous driving fields not only enable the conditional dynamics essential for quantum information processing, but also protect both qubits from dephasing during the gate operation.

Emulating Many-Body Localization with a Superconducting Quantum Processor

The law of statistical physics dictates that generic closed quantum many-body systems initialized in nonequilibrium will thermalize under their own dynamics. However, the emergence of many-body localization (MBL) owing to the interplay between interaction and disorder, which is in stark contrast to Anderson localization, which only addresses noninteracting particles in the presence of disorder, greatly challenges this concept, because it prevents the systems from evolving to the ergodic thermalized state. One critical evidence of MBL is the long-time logarithmic growth of entanglement entropy, and a direct observation of it is still elusive due to the experimental challenges in multiqubit single-shot measurement and quantum state tomography. Here we present an experiment fully emulating the MBL dynamics with a 10-qubit superconducting quantum processor, which represents a spin-1/2 XY model featuring programmable disorder and long-range spin-spin interactions. We provide essential signatures of MBL, such as the imbalance due to the initial nonequilibrium, the violation of eigenstate thermalization hypothesis, and, more importantly, the direct evidence of the long-time logarithmic growth of entanglement entropy. Our results lay solid foundations for precisely simulating the intriguing physics of quantum many-body systems on the platform of large-scale multiqubit superconducting quantum processors.

10-Qubit Entanglement and Parallel Logic Operations with a Superconducting Circuit

Here we report on the production and tomography of genuinely entangled Greenberger-Horne-Zeilinger states with up to ten qubits connecting to a bus resonator in a superconducting circuit, where the resonator-mediated qubit-qubit interactions are used to controllably entangle multiple qubits and to operate on different pairs of qubits in parallel. The resulting 10-qubit density matrix is probed by quantum state tomography, with a fidelity of 0.668±0.025. Our results demonstrate the largest entanglement created so far in solid-state architectures and pave the way to large-scale quantum computation.