We introduce a classical estimator for the post-processing of quantum phase estimation data generated either by quantum-Fourier-transform-based or quantum-signal-processing-based methods. We focus on the estimation of a single target phase promised to be within an interval where no other phases are present, which is typical of e.g. ground state energy estimation of gapped quantum systems. This allows us to perform phase estimation by filtering the signal within the promise region and recovering the phase through a moment-projection estimator. We show that our methods are robust in the presence of both additional phases outside the promise region and global depolarizing noise. In the noiseless case our estimator can achieve an exponential suppression of bias with respect to a naive mean estimator. In the presence of global depolarizing noise our estimator achieves a bias exponentially small in the circuit depth t at fixed circuit fidelity F, and a variance proportional to t^-2, improving by a factor of t^2 over the naive shifted-and-rescaled-mean approach. To mitigate realistic circuit-level noise, we combine our method with the explicit unbiasing scheme described in [Dutkiewicz et al., 2025]. As an illustrative example, we implement these estimators on a small-scale simulation of the Ising model, validating our theoretical results and finding better-than-expected performance for a global depolarizing noise approximation. The robustness of the moment-projection estimator in the presence of both multiple eigenvalues and realistic noise makes phase estimation with limited depth practical for early fault tolerant quantum experiments.

Simulating quantum dynamics is one of the most important applications of quantum computers. Traditional approaches for quantum simulation involve preparing the full evolved state of the system and then measuring some physical quantity. Here, we present a different and novel approach to quantum simulation that uses a compressed quantum state that we call the "shadow state". The amplitudes of this shadow state are proportional to the time-dependent expectations of a specific set of operators of interest, and it evolves according to its own Schrödinger equation. This evolution can be simulated on a quantum computer efficiently under broad conditions. Applications of this approach to quantum simulation problems include simulating the dynamics of exponentially large systems of free fermions or free bosons, the latter example recovering a recent algorithm for simulating exponentially many classical harmonic oscillators. These simulations are hard for classical methods and also for traditional quantum approaches, as preparing the full states would require exponential resources. Shadow Hamiltonian simulation can also be extended to simulate expectations of more complex operators such as two-time correlators or Green's functions, and to study the evolution of operators themselves in the Heisenberg picture.

As fully fault-tolerant quantum computers capable of solving useful problems remain a distant goal, we anticipate an era of early fault tolerance during which limited error correction is available. We propose a framework for designing early fault-tolerant algorithms by trading between error-correction overhead and residual logical noise, and apply it to quantum phase estimation (QPE). We develop a quantum-Fouriertransform-(QFT) based QPE technique that is robust to global depolarizing noise and outperforms the previous state of the art at low and moderate noise rates. We further introduce the explicitly unbiased maximum-likelihood estimator (EUMLE), a data-processing technique that mitigates arbitrary errors in QFT-based QPE schemes. EUMLE provides consistent asymptotically normal error-mitigated estimates, addressing the open problem of extending error mitigation beyond expectation-value estimation. Applying this scheme to the ground-state problem of the two-dimensional Hubbard model and various molecular Hamiltonians, we find that we can roughly halve the number of physical qubits with a similar to 10 times wall-clock time overhead, but that further reduction causes a steep run-time increase. This work provides an end-to-end analysis of early fault-tolerance cost reductions and space-time trade-offs, and identifies which areas can be improved in the future.

Quantum observables in the form of few-point correlators are the key to characterizing the dynamics of quantum many-body systems. In dynamics with fast entanglement generation, quantum observables generally become insensitive to the details of the underlying dynamics at long times due to the effects of scrambling. In experimental systems, repeated time-reversal protocols have been successfully implemented to restore sensitivities of quantum observables. Using a 103-qubit superconducting quantum processor, we characterize ergodic dynamics using the second-order out-of-time-order correlators, OTOC^(2). In contrast to dynamics without time reversal, OTOC^(2) are observed to remain sensitive to the underlying dynamics at long time scales. Furthermore, by inserting Pauli operators during quantum evolution and randomizing the phases of Pauli strings in the Heisenberg picture, we observe substantial changes in OTOC^(2) values. This indicates that OTOC^(2) is dominated by constructive interference between Pauli strings that form large loops in configuration space. The observed interference mechanism endows OTOC^(2) with a high degree of classical simulation complexity, which culminates in a set of large-scale OTOC^(2) measurements exceeding the simulation capacity of known classical algorithms. Further supported by an example of Hamiltonian learning through OTOC^(2), our results indicate a viable path to practical quantum advantage.

This note presents a simplified version of the OTOC^(2) problem that was recently experimentally implemented by Google Quantum AI and collaborators. We present a formulation of the problem for growing input size and hope this spurs further theoretical work on the problem.

We study the problem of learning the Hamiltonian of a many-body quantum system from experimental data. We show that the rate of learning depends on the amount of control available during the experiment. We consider three control models: one where time evolution can be augmented with instantaneous quantum operations, one where the Hamiltonian itself can be augmented by adding constant terms, and one where the experimentalist has no control over the system's time evolution. With continuous quantum control, we provide an adaptive algorithm for learning a many-body Hamiltonian at the Heisenberg limit: $T = \mathcal{O}(\epsilon^{-1})$, where $T$ is the total amount of time evolution across all experiments and $\epsilon$ is the target precision. This requires only preparation of product states, time-evolution, and measurement in a product basis. In the absence of quantum control, we prove that learning is standard quantum limited, $T = \Omega(\epsilon^{-2})$, for large classes of many-body Hamiltonians, including any Hamiltonian that thermalizes via the eigenstate thermalization hypothesis. These results establish a quadratic advantage in experimental runtime for learning with quantum control.

Conical intersections are topologically protected crossings between the potential energy surfaces of a molecular Hamiltonian, known to play an important role in chemical processes such as photoisomerization and non-radiative relaxation. They are characterized by a non-zero Berry phase, which is a topological invariant defined on a closed path in atomic coordinate space, taking the value π when the path encircles the intersection manifold. In this work, we show that for real molecular Hamiltonians, the Berry phase can be obtained by tracing a local optimum of a variational ansatz along the chosen path and estimating the overlap between the initial and final state with a control-free Hadamard test. Moreover, by discretizing the path into N points, we can use N single Newton-Raphson steps to update our state non-variationally. Finally, since the Berry phase can only take two discrete values (0 or π), our procedure succeeds even for a cumulative error bounded by a constant; this allows us to bound the total sampling cost and to readily verify the success of the procedure. We demonstrate numerically the application of our algorithm on small toy models of the formaldimine molecule (H2C=NH).

Quantum simulation is one of the most promising scientific applications of quantum computers. Due to decoherence and noise in current devices, it is however challenging to perform digital quantum simulation in a regime that is intractable with classical computers. In this work, we propose an experimental protocol for probing dynamics and equilibrium properties on near-term digital quantum computers. As a key ingredient of our work, we show that it is possible to study thermalization even with a relatively coarse Trotter decomposition of the Hamiltonian evolution of interest. Even though the step size is too large to permit a rigorous bound on the Trotter error, we observe that the system prethermalizes in accordance with previous results for Floquet systems. The dynamics closely resemble the thermalization of the model underlying the Trotterization up to long times. We extend the reach of our approach by developing an error mitigation scheme based on measurement and rescaling of survival probabilities. To demonstrate the effectiveness of the entire protocol, we apply it to the two-dimensional XY model and numerically verify its performance with realistic noise parameters for superconducting quantum devices. Our proposal thus provides a route to achieving quantum advantage for relevant problems in condensed matter physics.

For quantum computers to successfully solve real-world problems, it is necessary to tackle the challenge of noise: the errors which occur in elementary physical components due to unwanted or imperfect interactions. The theory of quantum fault tolerance can provide an answer in the long term, but in the coming era of `NISQ' machines we must seek to mitigate errors rather than completely remove them. This review surveys the diverse methods that have been proposed for quantum error mitigation, assesses their in-principle efficacy, and then describes the hardware demonstrations achieved to date. We identify the commonalities and limitations among the methods, noting how mitigation methods can be chosen according to the primary type of noise present, including algorithmic errors. Open problems in the field are identified and we discuss the prospects for realising mitigation-based devices that can deliver quantum advantage with an impact on science and business.

We consider a quantum computation that only extracts one bit of information per $N$-qubit quantum state preparation. This is relevant for error mitigation schemes where the remainder of the system is measured to detect errors. We optimize the estimation of the expectation value of an operator by its linear decomposition into bitwise-measurable terms. We prove that optimal decompositions must be in terms of reflections with eigenvalues $\pm1$. We find the optimal reflection decomposition of a fast-forwardable operator, and show a numerical improvement over a simple Pauli decomposition by a factor $N^{0.7}$

Variational quantum algorithms (VQAs) offer a promising path toward using near-term quantum hardware for applications in academic and industrial research. These algorithms aim to find approximate solutions to quantum problems by optimizing a parametrized quantum circuit using a classical optimization algorithm. A successful VQA requires fast and reliable classical optimization algorithms. Understanding and optimizing how off-the-shelf optimization methods perform in this context is important for the future of the field. In this work, we study the performance of four commonly used gradient-free optimization methods: SLSQP, COBYLA, CMA-ES, and SPSA, at finding ground-state energies of a range of small chemistry and material science problems. We test a telescoping sampling scheme (where the accuracy of the cost-function estimate provided to the optimizer is increased as the optimization converges) on all methods, demonstrating mixed results across our range of optimizers and problems chosen. We further hyperparameter tune two of the four optimizers (CMA-ES and SPSA) across a large range of models and demonstrate that with appropriate hyperparameter tuning, CMA-ES is competitive with and sometimes outperforms SPSA (which is not observed in the absence of hyperparameter tuning). Finally, we investigate the ability of an optimizer to beat the `sampling noise floor' given by the sampling noise on each cost-function estimate provided to the optimizer. Our results demonstrate the necessity for tailoring and hyperparameter-tuning known optimization techniques for inherently-noisy variational quantum algorithms and that the variational landscape that one finds in a VQA is highly problem- and system-dependent. This provides guidance for future implementations of these algorithms in the experiment.

Hamiltonian simulation is believed to be one of the first tasks where quantum computers can yield a quantum advantage. One of the most popular methods of Hamiltonian simulation is Trotterization, which makes use of the approximation $e^{i\sum_jA_j}\sim \prod_je^{iA_j}$ and higher-order corrections thereto. However, this leaves open the question of the order of operations (i.e. the order of the product over $j$, which is known to affect the quality of approximation). In some cases this order is fixed by the desire to minimise the error of approximation; when it is not the case, we propose that the order can be chosen to optimize compilation to a native quantum architecture. This presents a new compilation problem -- order-agnostic quantum circuit compilation -- which we prove is NP-hard in the worst case. In lieu of an easily-computable exact solution, we turn to methods of heuristic optimization of compilation. We focus on reinforcement learning due to the sequential nature of the compilation task, comparing it to simulated annealing and Monte Carlo tree search. While two of the methods outperform a naive heuristic, reinforcement learning clearly outperforms all others, with a gain of around 12% with respect to the second-best method and of around 50% compared to the naive heuristic in terms of the gate count. We further test the ability of RL to generalize across instances of the compilation problem, and find that a single learner is able to solve entire problem families. This demonstrates the ability of machine learning techniques to provide assistance in an order-agnostic quantum compilation task.

The famous, yet unsolved, Fermi-Hubbard model for strongly-correlated electronic systems is a prominent target for quantum computers. However, accurately representing the Fermi-Hubbard ground state for large instances may be beyond the reach of near-term quantum hardware. Here we show experimentally that an efficient, low-depth variational quantum algorithm with few parameters can reproduce important qualitative features of medium-size instances of the Fermi-Hubbard model. We address 1x8 and 2x4 instances on 16 qubits on a superconducting quantum processor, substantially larger than previous work based on less scalable compression techniques, and going beyond the family of 1D Fermi-Hubbard instances, which are solvable classically. Consistent with predictions for the ground state, we observe the onset of the metal-insulator transition and Friedel oscillations in 1D, and antiferromagnetic order in both 1D and 2D. We use a variety of error-mitigation techniques, including symmetries of the Fermi-Hubbard model and a recently developed technique tailored to simulating fermionic systems. We also introduce a new variational optimisation algorithm based on iterative Bayesian updates of a local surrogate model. Our scalable approach is a first step to using near-term quantum computers to determine low-energy properties of strongly-correlated electronic systems that cannot be solved exactly by classical computers.

We propose a quantum algorithm for inferring the molecular nuclear spin Hamiltonian from time-resolved measurements of spin-spin correlators, which can be obtained via nuclear magnetic resonance (NMR). We focus on learning the anisotropic dipolar term of the Hamiltonian, which generates dynamics that are challenging to classically simulate in some contexts. We demonstrate the ability to directly estimate the Jacobian and Hessian of the corresponding learning problem on a quantum computer, allowing us to learn the Hamiltonian parameters. We develop algorithms for performing this computation on both noisy near-term and future fault-tolerant quantum computers. We argue that the former is promising as an early beyond-classical quantum application since it only requires evolution of a local spin Hamiltonian. We investigate the example of a protein (ubiquitin) confined on a membrane as a benchmark of our method. We isolate small spin clusters, demonstrate the convergence of our learning algorithm on one such example, and then investigate the learnability of these clusters as we cross the ergodic to non-ergodic phase transition by suppressing the dipolar interaction. We see a clear correspondence between a drop in the multifractal dimension measured across many-body eigenstates of these clusters, and a transition in the structure of the Hessian of the learning cost function (from degenerate to learnable). Our hope is that such quantum computations might enable the interpretation and development of new NMR techniques for analyzing molecular structure.

Many quantum algorithms involve the evaluation of expectation values. Optimal strategies for estimating a single expectation value are known, requiring a number of state preparations that scales with the target error ϵ as O(1/ϵ). In this Letter, we address the task of estimating the expectation values of M different observables, each to within additive error ϵ, with the same 1/ϵ dependence. We describe an approach that leverages Gilyén et al.'s quantum gradient estimation algorithm to achieve O(sqrt[M]/ϵ) scaling up to logarithmic factors, regardless of the commutation properties of the M observables. We prove that this scaling is worst-case optimal in the high-precision regime if the state preparation is treated as a black box, even when the operators are mutually commuting. We highlight the flexibility of our approach by presenting several generalizations, including a strategy for accelerating the estimation of a collection of dynamic correlation functions.

While most work on the quantum simulation of chemistry has focused on computing energy surfaces, a similarly important application requiring subtly different algorithms is the computation of energy derivatives. Almost all molecular properties can be expressed an energy derivative, including molecular forces, which are essential for applications such as molecular dynamics simulations. Here, we introduce new quantum algorithms for computing molecular energy derivatives with significantly lower complexity than prior methods. Under cost models appropriate for noisy-intermediate scale quantum devices we demonstrate how low rank factorizations and other tomography schemes can be optimized for energy derivative calculations. We perform numerics revealing that our techniques reduce the number of circuit repetitions required by many orders of magnitude for even modest systems. In the context of fault-tolerant algorithms, we develop new methods of estimating energy derivatives with Heisenberg limited scaling incorporating state-of-the-art techniques for block encoding fermionic operators. Our results suggest that the calculation of forces on a single nucleus may be of similar cost to estimating energies of chemical systems, but that further developments are needed for quantum computers to meaningfully assist with molecular dynamics simulations.

Quantum phase estimation is a cornerstone in quantum algorithm design, allowing for the inference of eigenvalues of exponentially-large sparse matrices. The maximum rate at which these eigenvalues may be learned, –known as the Heisenberg limit–, is constrained by bounds on the circuit depth required to simulate an arbitrary Hamiltonian. Single-control qubit variants of quantum phase estimation have garnered interest in recent years due to lower circuit depth and minimal qubit overhead. In this work we show that these methods can achieve the Heisenberg limit, also when one is unable to prepare eigenstates of the system. Given a quantum subroutine which provides samples of a ‘phase function’ g(k) = ∑j Ajej with unknown eigenvalue phases φj and probabilities Aj at quantum cost O(k), we show how to estimate the phases {φj} with accuracy (root-mean-square) error δ for total quantum cost T = O(δ−1). Our scheme combines the idea of Heisenberg-limited multiorder quantum phase estimation for a single eigenvalue phase [1, 2] with subroutines with so-called dense quantum phase estimation which uses classical processing via time-series analysis for the QEEP problem [3] or the matrix pencil method. For our algorithm which adaptively fixes the choice for k in g(k) we prove Heisenberglimited scaling when we use the timeseries/QEEP subroutine. We present numerical evidence that using the matrix pencil technique the algorithm can achieve Heisenberg-limited scaling as well.

We optimise a translationally invariant, sequential quantum circuit on a superconducting quantum device to simulate the groundstate of the quantum Ising model through its quantum critical point. We further demonstrate how the dynamical quantum critical point found in quenches of this model across its quantum critical point can be simulated. Our approach avoids finite-size scaling effects by using sequential quantum circuits inspired by infinite matrix product states. We provide efficient circuits and a variety of error mitigation strategies to implement, optimise and time-evolve these states.

Two-qubit gates are important components of quantum computing. However, unwanted interactions between qubits (so-called parasitic gates) can be particularly problematic and degrade the performance of quantum applications. In this work, we present two software methods to mitigate parasitic two-qubit gate errors. The first approach is built upon the Cartan's KAK decomposition and keeps the original unitary decomposition for the error-free native two-qubit gate. It counteracts a parasitic two-qubit gate by only applying single-qubit rotations and therefore has no two-qubit gate overhead. We show the optimal choice of single-qubit mitigation gates. The second approach applies a numerical optimisation algorithm to re-compile a target unitary into the error-parasitic two-qubit gate plus single-qubit gates. We demonstrate these approaches on the CPhase-parasitic iSWAP-like gates. The KAK-based approach helps decrease unitary infidelity by a factor of 3 compared to the noisy implementation without error mitigation. When arbitrary single-qubit rotations are allowed, recompilation could completely mitigate the effect of parasitic errors but may require more native gates than the KAK-based approach. We also compare their average gate fidelity under realistic noise models, including relaxation and depolarising errors. Numerical results suggest that different approaches are advantageous in different error regimes, providing error mitigation guidance for near-term quantum computers.

Using 47 superconducting qubits, we implement the kicked Ising model which exhibits Majorana edge modes (MEMs) protected by Z 2 parity symmetry. Remarkably, we find that any multi-qubit Pauli operator overlapping with the MEMs exhibits a uniform decay rate comparable to single-qubit relaxation rates, irrespective of its size or composition. This finding allows us to accurately reconstruct the exponentially localized spatial profiles of the MEMs. Spectroscopic measurements further indicate exponentially suppressed hybridization between the MEMs at larger system sizes, which manifests as a strong resilience against low-frequency noise. Our work elucidates the noise sensitivity of symmetry-protected edge modes in a solid-state environment.