Near-term Quantum Research Articles

Hybrid Hamiltonian Simulation for Excitation Dynamics.

Hamiltonian simulation is one of the most anticipated applications of quantum computing. Quantum circuit depth for implementing Hamiltonian simulation is commonly time dependent using Trotter-Suzuki product formulas so that long time quantum dynamic simulations (QDSs) become impratical for near-term quantum processors. Hamiltonian simulation based on Cartan decomposition (CD) provides an appealing scheme for QDSs with fixed-depth circuits, while it is limited to a time-independent Hamiltonian. In this work, we generalize this CD-based Hamiltonian simulation algorithm for studying time-dependent systems by combining it with variational quantum algorithms. The time-dependent and time-independent parts of the Hamiltonian are treated by using variational and CD-based Hamiltonian simulation algorithms, respectively. As such, this hybrid Hamiltonian simulation requires only fixed-depth quantum circuits to handle time-dependent cases while maintaining a high accuracy. We apply this new algorithm to study the response of spin and molecular systems to δ-kick electric fields and obtain accurate spectra for these excitation processes.

Robust projective measurements through measuring code-inspired observables

Quantum measurements are ubiquitous in quantum information processing tasks, but errors can render their outputs unreliable. Here, we present a scheme that implements a robust projective measurement through measuring code-inspired observables. Namely, given a projective POVM, a classical code, and a constraint on the number of measurement outcomes each observable can have, we construct commuting observables whose measurement is equivalent to the projective measurement in the noiseless setting. Moreover, we can correct t errors on the classical outcomes of the observables’ measurement if the classical code corrects t errors. Since our scheme does not require the encoding of quantum data onto a quantum error correction code, it can help construct robust measurements for near-term quantum algorithms that do not use quantum error correction. Moreover, our scheme works for any projective POVM, and hence can allow robust syndrome extraction procedures in non-stabilizer quantum error correction codes.

Wigner state and process tomography on near-term quantum devices

We present an experimental scanning-based tomography approach for near-term quantum devices. The underlying method has previously been introduced in an ensemble-based NMR setting. Here we provide a tutorial-style explanation along with suitable software tools to guide experimentalists in its adaptation to near-term pure-state quantum devices. The approach is based on a Wigner-type representation of quantum states and operators. These representations provide a rich visualization of quantum operators using shapes assembled from a linear combination of spherical harmonics. These shapes (called droplets in the following) can be experimentally tomographed by measuring the expectation values of rotated axial tensor operators. We present an experimental framework for implementing the scanning-based tomography technique for circuit-based quantum computers and showcase results from IBM quantum experience. We also present a method for estimating the density and process matrices from experimentally tomographed Wigner functions (droplets). This tomography approach can be directly implemented using the Python-based software package DROPStomo.

Random Natural Gradient

Hybrid quantum-classical algorithms appear to be the most promising approach for near-term quantum applications. An important bottleneck is the classical optimization loop, where the multiple local minima and the emergence of barren plateaux make these approaches less appealing. To improve the optimization the Quantum Natural Gradient (QNG) method \cite{stokes2020quantum} was introduced – a method that uses information about the local geometry of the quantum state-space. While the QNG-based optimization is promising, in each step it requires more quantum resources, since to compute the QNG one requires O(m2) quantum state preparations, where m is the number of parameters in the parameterized circuit. In this work we propose two methods that reduce the resources/state preparations required for QNG, while keeping the advantages and performance of the QNG-based optimization. Specifically, we first introduce the Random Natural Gradient (RNG) that uses random measurements and the classical Fisher information matrix (as opposed to the quantum Fisher information used in QNG). The essential quantum resources reduce to linear O(m) and thus offer a quadratic "speed-up", while in our numerical simulations it matches QNG in terms of accuracy. We give some theoretical arguments for RNG and then benchmark the method with the QNG on both classical and quantum problems. Secondly, inspired by stochastic-coordinate methods, we propose a novel approximation to the QNG which we call Stochastic-Coordinate Quantum Natural Gradient that optimizes only a small (randomly sampled) fraction of the total parameters at each iteration. This method also performs equally well in our benchmarks, while it uses fewer resources than the QNG.

Infidelity Analysis of Digital Counter-Diabatic Driving in Simple Two-Qubit System.

Digitized counter-diabatic (CD) optimization algorithms have been proposed and extensively studied to enhance performance in quantum computing by accelerating adiabatic processes while minimizing energy transitions. While adding approximate counter-diabatic terms can initially introduce adiabatic errors that decrease over time, Trotter errors from decomposition approximation persist. On the other hand, increasing the high-order nested commutators for CD terms may improve adiabatic errors but could also introduce additional Trotter errors. In this article, we examine the two-qubit model to explore the interplay between approximate CD, adiabatic errors, Trotter errors, coefficients, and commutators. Through these analyses, we aim to gain insights into optimizing these factors for better fidelity, a shallower circuit depth, and a reduced gate number in near-term gate-based quantum computing.

Doubly Optimal Parallel Wire Cutting without Ancilla Qubits

A restriction in the quality and quantity of available qubits presents a substantial obstacle to the application of near-term and early fault-tolerant quantum computers in practical tasks. To confront this challenge, some techniques for effectively augmenting the system size through classical processing have been proposed; one promising approach is quantum circuit cutting. The main idea of quantum circuit cutting is to decompose an original circuit into smaller subcircuits and combine outputs from these subcircuits to recover the original output. Although this approach enables us to simulate larger quantum circuits beyond physically available circuits, it needs classical overheads quantified by two metrics: the sampling overhead in the number of measurements to reconstruct the original output, and the number of channels in the decomposition. Thus, it is crucial to devise a decomposition method that minimizes both of these metrics, thereby reducing the overall execution time. This paper studies the problem of decomposing the n-qubit identity channel, i.e., n-parallel wire cutting, into a set of local operations and classical communication; then we give an optimal wire-cutting method composed of channels based on mutually unbiased bases that achieves minimal overheads in both the sampling overhead and the number of channels, without ancilla qubits. This is in stark contrast to the existing method that achieves the optimal sampling overhead yet with ancilla qubits. Moreover, we derive a tight lower bound on the number of channels in parallel wire cutting without ancilla systems and show that only our method achieves this lower bound among the existing methods. Notably, our method shows an exponential improvement in the number of channels, compared to the aforementioned ancilla-assisted method that achieves optimal sampling overhead. Our work significantly alleviates the additional overheads for the wire-cutting method and may provide essential components for the early success of quantum computing. Published by the American Physical Society 2024

Towards determining the presence of barren plateaus in some chemically inspired variational quantum algorithms

In quantum chemistry, the variational quantum eigensolver (VQE) is a promising algorithm for molecular simulations on near-term quantum computers. However, VQEs using hardware-efficient circuits face scaling challenges due to the barren plateau problem. This raises the question of whether chemically inspired circuits from unitary coupled cluster (UCC) methods can avoid this issue. Here we provide theoretical evidence indicating they may not. By examining alternated dUCC ansätzes and relaxed Trotterized UCC ansätzes, we find that in the infinite depth limit, a separation occurs between particle-hole one- and two-body unitary operators. While one-body terms yield a polynomially concentrated energy landscape, adding two-body terms leads to exponential concentration. Numerical simulations support these findings, suggesting that popular 1-step Trotterized unitary coupled-cluster with singles and doubles (UCCSD) ansätze may not scale. Our results emphasize the link between trainability and circuit expressiveness, raising doubts about VQEs’ ability to surpass classical methods.

Optimization of Decoder Priors for Accurate Quantum Error Correction.

Accurate decoding of quantum error-correcting codes is a crucial ingredient in protecting quantum information from decoherence. It requires characterizing the error channels corrupting the logical quantum state and providing this information as a prior to the decoder. We introduce a reinforcement learning inspired method for calibrating these priors that aims to minimize the logical error rate. Our method significantly improves the decoding accuracy in repetition and surface code memory experiments executed on Google's Sycamore processor, outperforming the leading decoder-agnostic method by 16% and 3.3%, respectively. This calibration approach will serve as an important tool for maximizing the performance of both near-term and future error-corrected quantum devices.

Phase transitions in random circuit sampling

Undesired coupling to the surrounding environment destroys long-range correlations in quantum processors and hinders coherent evolution in the nominally available computational space. This noise is an outstanding challenge when leveraging the computation power of near-term quantum processors1. It has been shown that benchmarking random circuit sampling with cross-entropy benchmarking can provide an estimate of the effective size of the Hilbert space coherently available2–8. Nevertheless, quantum algorithms’ outputs can be trivialized by noise, making them susceptible to classical computation spoofing. Here, by implementing an algorithm for random circuit sampling, we demonstrate experimentally that two phase transitions are observable with cross-entropy benchmarking, which we explain theoretically with a statistical model. The first is a dynamical transition as a function of the number of cycles and is the continuation of the anti-concentration point in the noiseless case. The second is a quantum phase transition controlled by the error per cycle; to identify it analytically and experimentally, we create a weak-link model, which allows us to vary the strength of the noise versus coherent evolution. Furthermore, by presenting a random circuit sampling experiment in the weak-noise phase with 67 qubits at 32 cycles, we demonstrate that the computational cost of our experiment is beyond the capabilities of existing classical supercomputers. Our experimental and theoretical work establishes the existence of transitions to a stable, computationally complex phase that is reachable with current quantum processors.

Circuit-noise-resilient virtual distillation

Quantum error mitigation (QEM) is vital for improving quantum algorithms’ accuracy on noisy near-term devices. A typical QEM method, called Virtual Distillation (VD), can suffer from imperfect implementation, potentially leading to worse outcomes than without mitigation. To address this, we introduce Circuit-Noise-Resilient Virtual Distillation (CNR-VD), which includes a calibration process using simple input states to enhance VD’s performance despite circuit noise, aiming to recover the results of an ideally conducted VD circuit. Simulations show that CNR-VD significantly mitigates noise-induced errors in VD circuits, boosting accuracy by up to tenfold over standard VD. It provides positive error mitigation even under high noise, where standard VD fails. Furthermore, our estimator’s versatility extends its utility beyond VD, enhancing outcomes in general Hadamard-Test circuits. The proposed CNR-VD significantly enhances the noise-resilience of VD, and thus is anticipated to elevate the performance of quantum algorithm implementations on near-term quantum devices.

Variational quantum imaginary time evolution for matrix product state Ansatz with tests on transcorrelated Hamiltonians.

The matrix product state (MPS) Ansatz offers a promising approach for finding the ground state of molecular Hamiltonians and solving quantum chemistry problems. Building on this concept, the proposed technique of quantum circuit MPS (QCMPS) enables the simulation of chemical systems using a relatively small number of qubits. In this study, we enhance the optimization performance of the QCMPS Ansatz by employing the variational quantum imaginary time evolution (VarQITE) approach. Guided by McLachlan's variational principle, the VarQITE method provides analytical metrics and gradients, resulting in improved convergence efficiency and robustness of the QCMPS. We validate these improvements numerically through simulations of H2, H4, and LiH molecules. In addition, given that VarQITE is applicable to non-Hermitian Hamiltonians, we evaluate its effectiveness in preparing the ground state of transcorrelated Hamiltonians. This approach yields energy estimates comparable to the complete basis set (CBS) limit while using even fewer qubits. In particular, we perform simulations of the beryllium atom and LiH molecule using only three qubits, maintaining high fidelity with the CBS ground state energy of these systems. This qubit reduction is achieved through the combined advantages of both the QCMPS Ansatz and transcorrelation. Our findings demonstrate the potential practicality of this quantum chemistry algorithm on near-term quantum devices.

Enabling large-scale and high-precision fluid simulations on near-term quantum computers

Quantum computational fluid dynamics (QCFD) offers a promising alternative to classical computational fluid dynamics (CFD) by leveraging quantum algorithms for higher efficiency. This paper introduces a comprehensive QCFD method, including an iterative method “Iterative-QLS” that suppresses error in quantum linear solver, and a subspace method to scale the solution to a larger size. We implement our method on a superconducting quantum computer, demonstrating successful simulations of steady Poiseuille flow and unsteady acoustic wave propagation. The Poiseuille flow simulation achieved a relative error of less than 0.2%, and the unsteady acoustic wave simulation solved a 5043-dimensional matrix. We emphasize the utilization of the quantum–classical hybrid approach in applications of near-term quantum computers. By adapting to quantum hardware constraints and offering scalable solutions for large-scale CFD problems, our method paves the way for practical applications of near-term quantum computers in computational science.

Photonic counterdiabatic quantum optimization algorithm

One of the key applications of near-term quantum computers has been the development of quantum optimization algorithms. However, these algorithms have largely been focused on qubit-based technologies. Here, we propose a hybrid quantum-classical approximate optimization algorithm for photonic quantum computing, specifically tailored for addressing continuous-variable optimization problems. Inspired by counterdiabatic protocols, our algorithm reduces the required quantum operations for optimization compared to adiabatic protocols. This reduction enables us to tackle non-convex continuous optimization within the near-term era of quantum computing. Through illustrative benchmarking, we show that our approach can outperform existing state-of-the-art hybrid adiabatic quantum algorithms in terms of convergence and implementability. Our algorithm offers a practical and accessible experimental realization, bypassing the need for high-order operations and overcoming experimental constraints. We conduct a proof-of-principle demonstration on Xanadu’s eight-mode nanophotonic quantum chip, successfully showcasing the feasibility and potential impact of the algorithm.

Adaptive Variational Quantum Computing Approaches for Green's Functions and Nonlinear Susceptibilities.

We present and benchmark quantum computing approaches for calculating real-time single-particle Green's functions and nonlinear susceptibilities of Hamiltonian systems. The approaches leverage adaptive variational quantum algorithms for state preparation and propagation. Using automatically generated compact circuits, the dynamical evolution is performed over sufficiently long times to achieve adequate frequency resolution of the response functions. We showcase accurate Green's function calculations using a statevector simulator on classical hardware for Fermi-Hubbard chains of 4 and 6 sites, with maximal ansatz circuit depths of 65 and 424 layers, respectively, and for the molecule LiH with a maximal ansatz circuit depth of 81 layers. Additionally, we consider an antiferromagnetic quantum spin-1 model that incorporates the Dzyaloshinskii-Moriya interaction to illustrate calculations of the third-order nonlinear susceptibilities, which can be measured in two-dimensional coherent spectroscopy experiments. These results demonstrate that real-time approaches using adaptive parametrized circuits to evaluate linear and nonlinear response functions can be feasible with near-term quantum processors.

Detecting and eliminating quantum noise of quantum measurements

Quantum measurements are crucial for extracting information from quantum systems, but they are error-prone due to hardware imperfections in near-term devices. Measurement errors can be mitigated through classical post-processing, based on the assumption of a classical noise model. However, the coherence of quantum measurements leads to unavoidable quantum noise that defies this assumption. In this work, we introduce a two-stage procedure to systematically tackle such quantum noise in measurements. The idea is intuitive: we first detect and then eliminate quantum noise. In the first stage, inspired by coherence witness in the resource theory of quantum coherence, we design an efficient method to detect quantum noise. It works by fitting the difference between two measurement statistics to the Fourier series, where the statistics are obtained using maximally coherent states with relative phase and maximally mixed states as inputs. The fitting coefficients quantitatively benchmark quantum noise. In the second stage, we design various methods to eliminate quantum noise, inspired by the Pauli twirling technique. They work by executing randomly sampled Pauli gates before the measurement device and conditionally flipping the measurement outcomes in such a way that the effective measurement device contains only classical noise. We numerically demonstrate the two-stage procedure’s feasibility on the Baidu Quantum Platform. Notably, the results reveal significant suppression of quantum noise in measurement devices and substantial enhancement in quantum computation accuracy. We highlight that the two-stage procedure complements existing measurement error mitigation techniques, and they together form a standard toolbox for manipulating measurement errors in near-term quantum devices.

Increasing the Measured Effective Quantum Volume with Zero Noise Extrapolation

Quantum volume is a full-stack benchmark for near-term quantum computers. It quantifies the largest size of a square circuit which can be executed on the target device with reasonable fidelity. Error mitigation is a set of techniques intended to remove the effects of noise present in the computation of noisy quantum computers when computing an expectation value of interest. Effective quantum volume is a proposed metric that applies error mitigation to the quantum volume protocol to evaluate the effectiveness not only of the target device but also of the error mitigation algorithm. Digital zero-noise extrapolation is an error mitigation technique that estimates the noiseless expectation value using circuit folding to amplify errors by known scale factors and then extrapolating computed expectation values to the zero-noise limit. Here we demonstrate that zero-noise extrapolation, with global and local unitary folding with fractional scale factors, in conjunction with dynamical decoupling, can increase the effective quantum volume over the vendor-measured quantum volume. Specifically, we measure the effective quantum volume of four IBM Quantum superconducting processor units, obtaining values that are larger than the vendor-measured quantum volume on each device. This is the first such increase reported.

Trotter error bounds and dynamic multi-product formulas for Hamiltonian simulation

Multi-product formulas (MPFs) are linear combinations of Trotter circuits offering high-quality simulation of Hamiltonian time evolution with fewer Trotter steps. Here we report two contributions aimed at making multi-product formulas more viable for near-term quantum simulations. First, we extend the theory of Trotter error with commutator scaling developed by Childs [A. M. Childs , ] to multi-product formulas. Our result implies that multi-product formulas can achieve a quadratic reduction of Trotter error in 1-norm (nuclear norm) on arbitrary time intervals compared with the regular product formulas without increasing the required circuit depth or qubit connectivity. The number of circuit repetitions grows only by a constant factor. Second, we introduce dynamic multi-product formulas with time-dependent coefficients chosen to minimize a certain efficiently computable proxy for the Trotter error. We use a minimax estimation method to make dynamic multi-product formulas robust to uncertainty from algorithmic errors, sampling, and hardware noise. We call this method the minimax MPF and we provide a rigorous bound on its error. Published by the American Physical Society 2024

Large-scale simulations of Floquet physics on near-term quantum computers

Periodically driven quantum systems exhibit a diverse set of phenomena but are more challenging to simulate than their equilibrium counterparts. Here, we introduce the Quantum High-Frequency Floquet Simulation (QHiFFS) algorithm as a method to simulate fast-driven quantum systems on quantum hardware. Central to QHiFFS is the concept of a kick operator which transforms the system into a basis where the dynamics is governed by a time-independent effective Hamiltonian. This allows prior methods for time-independent simulation to be lifted to simulate Floquet systems. We use the periodically driven biaxial next-nearest neighbor Ising (BNNNI) model, a natural test bed for quantum frustrated magnetism and criticality, as a case study to illustrate our algorithm. We implemented a 20-qubit simulation of the driven two-dimensional BNNNI model on Quantinuum’s trapped ion quantum computer. Our error analysis shows that QHiFFS exhibits not only a cubic advantage in driving frequency ω but also a linear advantage in simulation time t compared to Trotterization.

Solving Boolean Satisfiability Problems With The Quantum Approximate Optimization Algorithm

One of the most prominent application areas for quantum computers is solving hard constraint satisfaction and optimization problems. However, detailed analyses of the complexity of standard quantum algorithms have suggested that outperforming classical methods for these problems would require extremely large and powerful quantum computers. The quantum approximate optimization algorithm (QAOA) is designed for near-term quantum computers, yet previous work has shown strong limitations on the ability of QAOA to outperform classical algorithms for optimization problems. Here we instead apply QAOA to hard constraint satisfaction problems, where both classical and quantum algorithms are expected to require exponential time. We analytically characterize the average success probability of QAOA on a constraint satisfaction problem commonly studied using statistical physics methods: random k-SAT at the threshold for satisfiability, as the number of variables n goes to infinity. We complement these theoretical results with numerical experiments on the performance of QAOA for small n, which match the limiting theoretical bounds closely. We then compare QAOA with leading classical solvers. For random 8-SAT, we find that for more than 14 quantum circuit layers, QAOA achieves more efficient scaling than the highest-performance classical solver we tested, WalkSATlm. Our results suggest that near-term quantum algorithms for solving constraint satisfaction problems may outperform their classical counterparts. Published by the American Physical Society 2024

Improving Quantum Optimization Algorithms by Constraint Relaxation

Quantum optimization is a significant area of quantum computing research with anticipated near-term quantum advantages. Current quantum optimization algorithms, most of which are hybrid variational-Hamiltonian-based algorithms, struggle to present quantum devices due to noise and decoherence. Existing techniques attempt to mitigate these issues through employing different Hamiltonian encodings or Hamiltonian clause pruning, but they often rely on optimistic assumptions rather than a deep analysis of the problem structure. We demonstrate how to formulate the problem Hamiltonian for a quantum approximate optimization algorithm that satisfies all the requirements to correctly describe the considered tactical aircraft deconfliction problem, achieving higher probabilities for finding solutions compared to previous works. Our results indicate that constructing Hamiltonians from an unconventional, quantum-specific perspective with a high degree of entanglement results in a linear instead of exponential number of entanglement gates instead and superior performance compared to standard formulations. Specifically, we achieve a higher probability of finding feasible solutions: finding solutions in nine out of nine instances compared to standard Hamiltonian formulations and quadratic programming formulations known from quantum annealers, which only found solutions in seven out of nine instances. These findings suggest that there is substantial potential for further research in quantum Hamiltonian design and that gate-based approaches may offer superior optimization performance over quantum annealers in the future.