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In the span of four decades, quantum computation has evolved from an intellectual curiosity to a potentially realizable technology. Today, small-scale demonstrations have become possible for quantum algorithmic primitives on hundreds of physical qubits and proof-of-principle error-correction on a single logical qubit. Nevertheless, despite significant progress and excitement, the path toward a full-stack scalable technology is largely unknown. There are significant outstanding quantum hardware, fabrication, software architecture, and algorithmic challenges that are either unresolved or overlooked. These issues could seriously undermine the arrival of utility-scale quantum computers for the foreseeable future. Here, we provide a comprehensive review of these scaling challenges. We show how the road to scaling could be paved by adopting existing semiconductor technology to build much higher-quality qubits, employing system engineering approaches, and performing distributed quantum computation within heterogeneous high-performance computing infrastructures. These opportunities for research and development could unlock certain promising applications, in particular, efficient quantum simulation/learning of quantum data generated by natural or engineered quantum systems. To estimate the true cost of such promises, we provide a detailed resource and sensitivity analysis for classically hard quantum chemistry calculations on surface-code error-corrected quantum computers given current, target, and desired hardware specifications based on superconducting qubits, accounting for a realistic distribution of errors. Furthermore, we argue that, to tackle industry-scale classical optimization and machine learning problems in a cost-effective manner, distributed quantum-assisted probabilistic computing with custom-designed accelerators should be considered as a complementary path toward scalability., Comment: 50 pages, 42 figures

We propose a novel technique for optimizing a modular fault-tolerant quantum computing architecture, taking into account any desired space-time trade--offs between the number of physical qubits and the fault-tolerant execution time of a quantum algorithm. We consider a concept architecture comprising a dedicated zone as a multi-level magic state factory and a core processor for efficient logical operations, forming a supply chain network for production and consumption of magic states. Using a heuristic algorithm, we solve the multi-objective optimization problem of minimizing space and time subject to a user-defined error budget for the success of the computation, taking the performance of various fault-tolerant protocols such as quantum memory, state preparation, magic state distillation, code growth, and logical operations into account. As an application, we show that physical quantum resource estimation reduces to a simple model involving a small number of key parameters, namely, the circuit volume, the error prefactors ($\mu$) and error suppression rates ($\Lambda$) of the fault-tolerant protocols, and an allowed slowdown factor ($\beta$). We show that, in the proposed architecture, $10^5$--$10^8$ physical qubits are required for quantum algorithms with $T$-counts in the range $10^6$--$10^{15}$ and logical qubit counts in the range $10^2$--$10^4$, when run on quantum computers with quantum memory $\Lambda$ in the range 3--10, for all slowdown factors $\beta \geq 0.2$., Comment: 21 pages, 6 figures

Echoed conditional displacement (ECD) gates for bosonic systems have become the key element for real-time quantum error correction beyond the break-even point. These gates are characterized by a single complex parameter $\beta$, and can be constructed using Gaussian pulses and free evolutions with the help of an ancillary transmon qubit. We show that there is a lower bound for the gate time in the standard construction of an ECD gate. We present a method for optimizing the pulse shape of an ECD gate using a pulse-shaping technique subject to a set of experimental constraints. Our optimized pulse shapes remain symmetric, and can be applied to a range of target values of $\beta$ by tuning only the amplitude. We demonstrate that the total gate time of an ECD gate for a small value of $\beta$ can be reduced either by relaxing the no-overlap constraint on the primitives used in the standard construction or via our optimal-control method. We show a slight advantage of the optimal-control method by demonstrating a reduction in the preparation time of a $|+Z_\mathrm{GKP}>$ logical state by $\thicksim$$10\%$.

Fault-tolerant quantum computation using two-dimensional topological quantum error correcting codes can benefit from multi-qubit long-range operations. By using simple commutation rules, a quantum circuit can be transpiled into a sequence of solely non-Clifford multi-qubit gates. Prior work on fault-tolerant compilation avoids optimal scheduling of such gates since they reduce the parallelizability of the circuit. We observe that the reduced parallelization potential is outweighed by the significant reduction in the number of gates. We therefore devise a method for scheduling multi-qubit lattice surgery using an earliest-available-first policy, solving the associated forest packing problem using a representation of the multi-qubit gates as Steiner trees. Our extensive testing on random and application-inspired circuits demonstrates the method's scalability and performance. We show that the transpilation significantly reduces the circuit length on the set of circuits tested, and that the resulting circuit of multi-qubit gates has a further reduction in the expected circuit execution time compared to serial execution., Comment: 23 pages, 7 figures, 4 tables

We investigate the advantages of using autoregressive neural quantum states as ansatze for classical shadow tomography to improve its predictive power. We introduce a novel estimator for optimizing the cross-entropy loss function using classical shadows, and a new importance sampling strategy for estimating the loss gradient during training using stabilizer samples collected from classical shadows. We show that this loss function can be used to achieve stable reconstruction of GHZ states using a transformer-based neural network trained on classical shadow measurements. This loss function also enables the training of neural quantum states representing purifications of mixed states. Our results show that the intrinsic capability of autoregressive models in representing physically well-defined density matrices allows us to overcome the weakness of Pauli-based classical shadow tomography in predicting both high-weight observables and nonlinear observables such as the purity of pure and mixed states., Comment: 15 pages, 5 figures

In the traditional approach to controlling superconducting qubits using microwave pulses, the field of pulse shaping has emerged in order to assist in the removal of leakage and increase gate fidelity. However, the challenge of scaling microwave control electronics has created an opportunity to explore alternative methods such as single-flux quantum (SFQ) pulses. For qubits controlled by SFQ pulses, high fidelity gates can be achieved by optimizing the binary control sequence. We extend the notion of the derivative removal by adiabatic gate (DRAG) framework a transmon qubit controlled by SFQ drivers and propose pulse sequences that can be stored in 22 bits or fewer, with gate fidelities exceeding 99.99%. This modest memory requirement could help reduce the footprint of the SFQ coprocessors and power dissipation while preserving their inherent advantages of scalability and cost-effectiveness.

Neural decoders for quantum error correction (QEC) rely on neural networks to classify syndromes extracted from error correction codes and find appropriate recovery operators to protect logical information against errors. Its ability to adapt to hardware noise and long-term drifts make neural decoders a promising candidate for inclusion in a fault-tolerant quantum architecture. However, given their limited scalability, it is prudent that small-scale (local) neural decoders are treated as first stages of multi-stage decoding schemes for fault-tolerant quantum computers with millions of qubits. In this case, minimizing the decoding time to match the stabilization measurements frequency and a tight co-integration with the QPUs is highly desired. Cryogenic realizations of neural decoders can not only improve the performance of higher stage decoders, but they can minimize communication delays, and alleviate wiring bottlenecks. In this work, we design and analyze a neural decoder based on an in-memory computation (IMC) architecture, where crossbar arrays of resistive memory devices are employed to both store the synaptic weights of the neural decoder and perform analog matrix-vector multiplications. In simulations supported by experimental measurements, we investigate the impact of TiOx-based memristive devices' non-idealities on decoding fidelity. We develop hardware-aware re-training methods to mitigate the fidelity loss, restoring the ideal decoder's pseudo-threshold for the distance-3 surface code. This work provides a pathway to scalable, fast, and low-power cryogenic IMC hardware for integrated fault-tolerant QEC.

We investigate whether long-run persistent chain Monte Carlo simulation of Langevin dynamics improves the quality of the representations achieved by energy-based models (EBM). We consider a scheme wherein Monte Carlo simulation of a diffusion process using a trained EBM is used to improve the adversarial robustness and the calibration score of an independent classifier network. Our results show that increasing the computational budget of Gibbs sampling in persistent contrastive divergence improves the calibration and adversarial robustness of the model, elucidating the practical merit of realizing new quantum and classical hardware and software for efficient Gibbs sampling from continuous energy potentials.

Quantum state tomography (QST) is the art of reconstructing an unknown quantum state through measurements. It is a key primitive for developing quantum technologies. Neural network quantum state tomography (NNQST), which aims to reconstruct the quantum state via a neural network ansatz, is often implemented via a basis-dependent cross-entropy loss function. State-of-the-art implementations of NNQST are often restricted to characterizing a particular subclass of states, to avoid an exponential growth in the number of required measurement settings. To provide a more broadly applicable method for efficient state reconstruction, we present "neural-shadow quantum state tomography" (NSQST)-an alternative neural network-based QST protocol that uses infidelity as the loss function. The infidelity is estimated using the classical shadows of the target state. Infidelity is a natural choice for training loss, benefiting from the proven measurement sample efficiency of the classical shadow formalism. Furthermore, NSQST is robust against various types of noise without any error mitigation. We numerically demonstrate the advantage of NSQST over NNQST at learning the relative phases of three target quantum states of practical interest, as well as the advantage over direct shadow estimation. NSQST greatly extends the practical reach of NNQST and provides a novel route to effective quantum state tomography., Comment: 20 pages, 14 figures. New subsection added and several other changes. Final version published in Physical Review Research

Gibbs sampling from continuous real-valued functions is a challenging problem of interest in machine learning. Here we leverage quantum Fourier transforms to build a quantum algorithm for this task when the function is periodic. We use the quantum algorithms for solving linear ordinary differential equations to solve the Fokker--Planck equation and prepare a quantum state encoding the Gibbs distribution. We show that the efficiency of interpolation and differentiation of these functions on a quantum computer depends on the rate of decay of the Fourier coefficients of the Fourier transform of the function. We view this property as a concentration of measure in the Fourier domain, and also provide functional analytic conditions for it. Our algorithm makes zeroeth order queries to a quantum oracle of the function. Despite suffering from an exponentially long mixing time, this algorithm allows for exponentially improved precision in sampling, and polynomial quantum speedups in mean estimation in the general case, and particularly under geometric conditions we identify for the critical points of the energy function., Comment: 50 pages, version accepted in ICML 2024 / PMLR

We investigate the possibility of solving continuous non-convex optimization problems using a network of interacting quantum optical oscillators. We propose a native encoding of continuous variables in analog signals associated with the quadrature operators of a set of quantum optical modes. Optical coupling of the modes and noise introduced by vacuum fluctuations from external reservoirs or by weak measurements of the modes are used to optically simulate a diffusion process on a set of continuous random variables. The process is run sufficiently long for it to relax into the steady state of an energy potential defined on a continuous domain. As a first demonstration, we numerically benchmark solving box-constrained quadratic programming (BoxQP) problems using these settings. We consider delay-line and measurement-feedback variants of the experiment. Our benchmarking results demonstrate that in both cases the optical network is capable of solving BoxQP problems over three orders of magnitude faster than a state-of-the-art classical heuristic., Comment: 10 pages, 6 figures

We propose a scheme for solving mixed-integer programming problems in which the optimization problem is translated to a ground-state preparation problem on a set of bosonic quantum field modes (qumodes). We perform numerical demonstrations by simulating a circuit-based optical quantum computer with each individual qumode prepared in a Gaussian state. We simulate an adiabatic evolution from an initial mixing Hamiltonian, written in terms of the momentum operators of the qumodes, to a final Hamiltonian which is a polynomial of the position and boson number operators. In these demonstrations, we solve a variety of small non-convex optimization problems in integer programming, continuous non-convex optimization, and mixed-integer programming., Comment: 13 pages, 9 figures

We propose a circuit-model quantum algorithm for eigenpath traversal that is based on a combination of concepts from Grover's search and adiabatic quantum computation. Our algorithm deploys a sequence of reflections determined from eigenspaces of instantaneous Hamiltonians defined along an adiabatic schedule in order to prepare a ground state of a target problem Hamiltonian. We provide numerical evidence suggesting that, for combinatorial search problems, our algorithm can find a solution faster, on average, than Grover's search. We demonstrate our findings by applying both algorithms to solving the NP-hard MAX-2SAT problem., Comment: 10 pages, 9 figures

We introduce a novel transpiler for the placement and routing of quantum circuits on arbitrary target hardware architectures. We use finite-horizon, and optionally discounted, reward functions to heuristically find a suitable placement and routing policy. We employ a finite lookahead to refine the reward functions when breaking a tie between multiple policies. We benchmark our transpiler against multiple alternative solutions and on various test sets of quantum algorithms to demonstrate the benefits of our approach., Comment: 11 pages, 7 figures, 3 tables

Near-term quantum computers provide a promising platform for finding ground states of quantum systems, which is an essential task in physics, chemistry, and materials science. Near-term approaches, however, are constrained by the effects of noise as well as the limited resources of near-term quantum hardware. We introduce "neural error mitigation," which uses neural networks to improve estimates of ground states and ground-state observables obtained using near-term quantum simulations. To demonstrate our method's broad applicability, we employ neural error mitigation to find the ground states of the H$_2$ and LiH molecular Hamiltonians, as well as the lattice Schwinger model, prepared via the variational quantum eigensolver (VQE). Our results show that neural error mitigation improves numerical and experimental VQE computations to yield low energy errors, high fidelities, and accurate estimations of more-complex observables like order parameters and entanglement entropy, without requiring additional quantum resources. Furthermore, neural error mitigation is agnostic with respect to the quantum state preparation algorithm used, the quantum hardware it is implemented on, and the particular noise channel affecting the experiment, contributing to its versatility as a tool for quantum simulation., Comment: 20 pages, 4 main figures, 7 supplementary figures, 1 supplementary table

We study the performance scaling of three quantum algorithms for combinatorial optimization: measurement-feedback coherent Ising machines (MFB-CIM), discrete adiabatic quantum computation (DAQC), and the D\"urr-Hoyer algorithm for quantum minimum finding (DH-QMF) that is based on Grover's search. We use MaxCut problems as our reference for comparison, and time-to-solution (TTS) as a practical measure of performance for these optimization algorithms. We empirically observe a $\Theta(2^{\sqrt{n}})$ scaling for the median TTS for MFB-CIM, in comparison to the exponential scaling with the exponent $n$ for DAQC and the provable $\widetilde{\mathcal O}\left(\sqrt{2^n}\right)$ scaling for DH-QMF. We conclude that these scaling complexities result in a dramatic performance advantage for MFB-CIM in comparison to the other two algorithms for solving MaxCut problems., Comment: 24 pages, 20 figures

Reinforcement learning (RL) has become a proven method for optimizing a procedure for which success has been defined, but the specific actions needed to achieve it have not. We apply the so-called "black box" method of RL to what has been referred as the "black art" of simulated annealing (SA), demonstrating that an RL agent based on proximal policy optimization can, through experience alone, arrive at a temperature schedule that surpasses the performance of standard heuristic temperature schedules for two classes of Hamiltonians. When the system is initialized at a cool temperature, the RL agent learns to heat the system to "melt" it, and then slowly cool it in an effort to anneal to the ground state; if the system is initialized at a high temperature, the algorithm immediately cools the system. We investigate the performance of our RL-driven SA agent in generalizing to all Hamiltonians of a specific class; when trained on random Hamiltonians of nearest-neighbour spin glasses, the RL agent is able to control the SA process for other Hamiltonians, reaching the ground state with a higher probability than a simple linear annealing schedule. Furthermore, the scaling performance (with respect to system size) of the RL approach is far more favourable, achieving a performance improvement of one order of magnitude on L=14x14 systems. We demonstrate the robustness of the RL approach when the system operates in a "destructive observation" mode, an allusion to a quantum system where measurements destroy the state of the system. The success of the RL agent could have far-reaching impact, from classical optimization, to quantum annealing, to the simulation of physical systems.

We discuss the problem of finite-horizon dynamic programming (DP) on a quantum computer. We introduce a query model for studying quantum and classical algorithms for solving DP problems, and provide example oracle constructions for the travelling salesperson problem, the minimum set-cover problem, and the edit distance problem. We formulate open questions regarding quadratic quantum speedups for DP and discuss their implications. We then prove lower bounds for the query complexity of quantum algorithms and classical randomized algorithms for DP, and show that no greater-than-quadratic speedup can be achieved for solving DP problems., Comment: We were not able to amend the proof of Theorem III.5 of the previous version (v2) of this manuscript. Therefore, we have hereby withdrawn the query complexity upper bound claims that were stated in our earlier submission

We introduce two quantum algorithms for solving structured prediction problems. We first show that a stochastic gradient descent that uses the quantum minimum finding algorithm and takes its probabilistic failure into account solves the structured prediction problem with a runtime that scales with the square root of the size of the label space, and in $\widetilde O\left(1/\epsilon\right)$ with respect to the precision, $\epsilon$, of the solution. Motivated by robust inference techniques in machine learning, we then introduce another quantum algorithm that solves a smooth approximation of the structured prediction problem with a similar quantum speedup in the size of the label space and a similar scaling in the precision parameter. In doing so, we analyze a variant of stochastic gradient descent for convex optimization in the presence of an additive error in the calculation of the gradients, and show that its convergence rate does not deteriorate if the additive errors are of the order $O(\sqrt\epsilon)$. This algorithm uses quantum Gibbs sampling at temperature $\Omega (\epsilon)$ as a subroutine. Based on these theoretical observations, we propose a method for using quantum Gibbs samplers to combine feedforward neural networks with probabilistic graphical models for quantum machine learning. Our numerical results using Monte Carlo simulations on an image tagging task demonstrate the benefit of the approach.

Finding efficient decoders for quantum error correcting codes adapted to realistic experimental noise in fault-tolerant devices represents a significant challenge. In this paper we introduce several decoding algorithms complemented by deep neural decoders and apply them to analyze several fault-tolerant error correction protocols such as the surface code as well as Steane and Knill error correction. Our methods require no knowledge of the underlying noise model afflicting the quantum device making them appealing for real-world experiments. Our analysis is based on a full circuit-level noise model. It considers both distance-three and five codes, and is performed near the codes pseudo-threshold regime. Training deep neural decoders in low noise rate regimes appears to be a challenging machine learning endeavour. We provide a detailed description of our neural network architectures and training methodology. We then discuss both the advantages and limitations of deep neural decoders. Lastly, we provide a rigorous analysis of the decoding runtime of trained deep neural decoders and compare our methods with anticipated gate times in future quantum devices. Given the broad applications of our decoding schemes, we believe that the methods presented in this paper could have practical applications for near term fault-tolerant experiments., Comment: 26 pages, 8 tables, 28 figures. Comments welcome! v2 includes fixes to minor typos and expanded discussions

Recent advancements in quantum annealing hardware and numerous studies in this area suggests that quantum annealers have the potential to be effective in solving unconstrained binary quadratic programming problems. Naturally, one may desire to expand the application domain of these machines to problems with general discrete variables. In this paper, we explore the possibility of employing quantum annealers to solve unconstrained quadratic programming problems over a bounded integer domain. We present an approach for encoding integer variables into binary ones, thereby representing unconstrained integer quadratic programming problems as unconstrained binary quadratic programming problems. To respect some of the limitations of the currently developed quantum annealers, we propose an integer encoding, named bounded- coefficient encoding, in which we limit the size of the coefficients that appear in the encoding. Furthermore, we propose an algorithm for finding the upper bound on the coefficients of the encoding using the precision of the machine and the coefficients of the original integer problem. Finally, we experimentally show that this approach is far more resilient to the noise of the quantum annealers compared to traditional approaches for the encoding of integers in base two.

Recent theoretical and experimental results suggest the possibility of using current and near-future quantum hardware in challenging sampling tasks. In this paper, we introduce free energy-based reinforcement learning (FERL) as an application of quantum hardware. We propose a method for processing a quantum annealer's measured qubit spin configurations in approximating the free energy of a quantum Boltzmann machine (QBM). We then apply this method to perform reinforcement learning on the grid-world problem using the D-Wave 2000Q quantum annealer. The experimental results show that our technique is a promising method for harnessing the power of quantum sampling in reinforcement learning tasks.

We investigate whether quantum annealers with select chip layouts can outperform classical computers in reinforcement learning tasks. We associate a transverse field Ising spin Hamiltonian with a layout of qubits similar to that of a deep Boltzmann machine (DBM) and use simulated quantum annealing (SQA) to numerically simulate quantum sampling from this system. We design a reinforcement learning algorithm in which the set of visible nodes representing the states and actions of an optimal policy are the first and last layers of the deep network. In absence of a transverse field, our simulations show that DBMs are trained more effectively than restricted Boltzmann machines (RBM) with the same number of nodes. We then develop a framework for training the network as a quantum Boltzmann machine (QBM) in the presence of a significant transverse field for reinforcement learning. This method also outperforms the reinforcement learning method that uses RBMs.

We view the inertia construction of algebraic stacks as an operator on the Grothendieck groups of various categories of algebraic stacks. We show that the inertia operator is locally finite and diagonalizable. This is proved for the Grothendieck group of Deligne-Mumford stacks over a base scheme and the category of quasi-split Artin stacks defined over a field of characteristic zero. Motivated by the quasi-splitness condition, in [2] we consider the inertia operator of the Hall algebra of algebroids, and applications of it in generalized Donaldson-Thomas theory., Comment: 39 pages

We study the action of the inertia operator on the motivic Hall algebra, and prove that it is diagonalizable. This leads to a filtration of the Hall algebra, whose associated graded algebra is commutative. In particular, the degree 1 subspace forms a Lie algebra, which we call the Lie algebra of virtually indecomposable elements, following Joyce. We prove that the integral of virtually indecomposable elements admits an Euler characteristic specialization. In order to take advantage of the fact that our inertia groups are unit groups in algebras, we introduce the notion of algebroid., Comment: Version 2: polished the paper a bit

An earlier work [18] proposes a method for solving the Lagrangian dual of a constrained binary quadratic programming problem via quantum adiabatic evolution using an outer approximation method. This should be an efficient prescription for solving the Lagrangian dual problem in the presence of an ideally noise-free quantum adiabatic system. However, current implementations of quantum annealing systems demand methods that are efficient at handling possible sources of noise. In this paper, we consider a subgradient method for finding an optimal primal-dual pair for the Lagrangian dual of a constrained binary polynomial programming problem. We then study the quadratic stable set (QSS) problem as a case study. We see that this method applied to the QSS problem can be viewed as an instance-dependent penalty-term approach that avoids large penalty coefficients. Finally, we report our experimental results of using the D-Wave 2X quantum annealer and conclude that our approach helps this quantum processor to succeed more often in solving these problems compared to the usual penalty-term approaches.

Quantum adiabatic evolution is perceived as useful for binary quadratic programming problems that are a priori unconstrained. For constrained problems, it is a common practice to relax linear equality constraints as penalty terms in the objective function. However, there has not yet been proposed a method for efficiently dealing with inequality constraints using the quantum adiabatic approach. In this paper, we give a method for solving the Lagrangian dual of a binary quadratic programming (BQP) problem in the presence of inequality constraints and employ this procedure within a branch-and-bound framework for constrained BQP (CBQP) problems., Comment: 20 pages, 2 figures