Finding the solution to linear ordinary differential equations of the form $\partial_t u(t) = -A(t)u(t)$ has been a promising theoretical avenue for \textit{asymptotic} quantum speedups. However, despite the improvements to existing quantum differential equation solvers over the years, little is known about \textit{constant factor} costs of such quantum algorithms. This makes it challenging to assess the prospects for using these algorithms in practice. In this work, we prove constant factor bounds for a promising new quantum differential equation solver, the linear combination of Hamiltonian simulation (LCHS) algorithm. Our bounds are formulated as the number of queries to a unitary $U_A$ that block encodes the generator $A$. In doing so, we make several algorithmic improvements such as tighter truncation and discretization bounds on the LCHS kernel integral, a more efficient quantum compilation scheme for the SELECT operator in LCHS, as well as use of a constant-factor bound for oblivious amplitude amplification, which may be of general interest. To the best of our knowledge, our new formulae improve over previous state of the art by at least two orders of magnitude, where the speedup can be far greater if state preparation has a significant cost. Accordingly, for any previous resource estimates of time-independent linear differential equations for the most general case whereby the dynamics are not \textit{fast-forwardable}, these findings provide a 110x reduction in runtime costs. This analysis contributes towards establishing more promising applications for quantum computing.

A major thrust in quantum algorithm development over the past decade has been the search for the quantum algorithms that will deliver practical quantum advantage first. Today's quantum computers and even early fault-tolerant quantum computers will be limited in the number of operations they can implement per circuit. We introduce quantum algorithms for ground state energy estimation (GSEE) that accommodate this design constraint. The first estimates ground state energies and has a quadratic improvement on the ground state overlap parameter compared to other methods in this regime. The second certifies that the estimated ground state energy is within a specified error tolerance of the true ground state energy, addressing the issue of gap estimation that beleaguers several ground state preparation and energy estimation algorithms. We note, however, that the scaling of this certification technique is, unfortunately, worse than that of the GSEE algorithm. These algorithms are based on a novel use of the quantum computer to facilitate rejection sampling. After a classical computer is used to draw samples, the quantum computer is used to accept or reject the samples. The set of accepted samples correspond to draws from a target distribution. While we use this technique for ground state energy estimation, it may find broader application. Our work pushes the boundaries of what operation-limited quantum computers are capable of and thus brings the target of quantum advantage closer to the present.

Over the past decade, research in quantum computing has tended to fall into one of two camps: near-term intermediate scale quantum (NISQ) and fault-tolerant quantum computing (FTQC). Yet, a growing body of work has been investigating how to use quantum computers in transition between these two eras. This envisions operating with tens of thousands to millions of physical qubits, able to support fault-tolerant protocols, though operating close to the fault-tolerant threshold. Two challenges emerge from this picture: how to model the performance of devices that are continually improving and how to design algorithms to make the most use of these devices? In this work we develop a model for the performance of early fault-tolerant quantum computing (EFTQC) architectures and use this model to elucidate the regimes in which algorithms suited to such architectures are advantageous. As a concrete example, we show that, for the canonical task of phase estimation, in a regime of moderate scalability and using just over one million physical qubits, the ``reach'' of the quantum computer can be extended (compared to the standard approach) from 90-qubit instances to over 130-qubit instances using a simple early fault-tolerant quantum algorithm, which reduces the number of operations per circuit by a factor of 100 and increases the number of circuit repetitions by a factor of 10,000. This clarifies the role that such algorithms might play in the era of limited-scalability quantum computing.

Understanding the physics of strongly correlated materials is one of the grand challenge problems for physics today. A large class of scientifically interesting materials, from high-T_c superconductors to spin liquids, involve medium to strong correlations, and building a holistic understanding of these materials is critical. Doing so is hindered by the competition between the kinetic energy and Coulomb repulsion, which renders both analytic and numerical methods unsatisfactory for describing interacting materials. Fault-tolerant quantum computers have been proposed as a path forward to overcome these difficulties, but this potential capability has not yet been fully assessed. Here, using the multi-orbital Fermi-Hubbard model as a representative model and a source of scalable problem specifications, we estimate the resource costs needed to use fault-tolerant quantum computers for obtaining experimentally relevant quantities such as correlation function estimation. We find that advances in quantum algorithms and hardware will be needed in order to reduce quantum resources and feasibly address utility-scale problem instances.

The hope of the quantum computing field is that quantum architectures are able to scale up and realize fault-tolerant quantum computing. Due to engineering challenges, such "cheap" error correction may be decades away. In the meantime, we anticipate an era of "costly" error correction, or early fault-tolerant quantum computing. Costly error correction might warrant settling for error-prone quantum computations. This motivates the development of quantum algorithms which are robust to some degree of error as well as methods to analyze their performance in the presence of error. We introduce a randomized algorithm for the task of phase estimation and give an analysis of its performance under two simple noise models. In both cases the analysis leads to a noise threshold, below which arbitrarily high accuracy can be achieved by increasing the number of samples used in the algorithm. As an application of this general analysis, we compute the maximum ratio of the largest circuit depth and the dephasing scale such that performance guarantees hold. We calculate that the randomized algorithm can succeed with arbitrarily high probability as long as the required circuit depth is less than 0.916 times the dephasing scale.

One promising field of quantum computation is the simulation of quantum systems, and specifically, the task of ground state energy estimation (GSEE). Ground state preparation (GSP) is a crucial component in GSEE algorithms, and classical methods like Hartree-Fock state preparation are commonly used. However, the efficiency of such classical methods diminishes exponentially with increasing system size in certain cases. In this study, we investigated whether in those cases quantum heuristic GSP methods could improve the overlap values compared to Hartree-Fock. Moreover, we carefully studied the performance gain for GSEE algorithms by exploring the trade-off between the overlap improvement and the associated resource cost in terms of T-gates of the GSP algorithm. Our findings indicate that quantum heuristic GSP can accelerate GSEE tasks, already for computationally affordable strongly-correlated systems of intermediate size. These results suggest that quantum heuristic GSP has the potential to significantly reduce the runtime requirements of GSEE algorithms, thereby enhancing their suitability for implementation on quantum hardware.

This study examines the potential for fault-tolerant quantum computers to provide utility in fluid dynamics simulations, with a focus on drag force calculations for ship hull design. We assess whether quantum algorithms can surpass classical computational limits by generating detailed quantum resource estimates (QREs) in terms of logical qubits and T-gate counts. Our analysis is based on a quantum algorithm leveraging Carleman linearization of the lattice Boltzmann method (LBM), which has been suggested to offer exponential speedup. We develop efficient block encodings for LBM matrices and a method for amplitude-encoding drag force. We apply the method to the simple case of fluid flow past a sphere across a range of Reynolds numbers (Re). We estimate the required (logical qubits)×(T-gates), finding them to be prohibitively large, ranging from 10^21 to 10^39. While classical simulations scale as O(Re^3), our QREs exhibit a modest polynomial scaling of O(Re^2.68), indicating no exponential quantum advantage. We attribute this limitation to an intrinsic power-law relationship between spatial grid resolution and time-stepping requirements that is a fundamental characteristic of explicit methods for evolving nonlinear differential equations. Thus, quantum computers are unlikely to provide utility in applications that require time-evolving fluids and other systems of nonlinear differential equations.

The industrial manufacturing of chemicals consumes a significant amount of energy and raw materials. In principle, the development of new catalysts could greatly improve the efficiency of chemical production. However, the discovery of viable catalysts can be exceedingly challenging because it is difficult to know the efficacy of a candidate without experimentally synthesizing and characterizing it. This study explores the feasibility of using fault-tolerant quantum computers to accelerate the discovery of homogeneous catalysts for nitrogen fixation, an industrially important chemical process. It introduces a set of ground-state energy estimation problems representative of calculations needed for the discovery of homogeneous catalysts and analyzes them on three dimensions: economic utility, classical hardness, and quantum resource requirements. For the highest utility problem considered, two steps of a catalytic cycle for the generation of cyanate anion from dinitrogen, the economic utility of running these computations is estimated to be $200,000, and the required runtime for double-factorized phase estimation on a fault-tolerant superconducting device is estimated under conservative assumptions to be 139,000 QPU-hours. The computational cost of an equivalent DMRG calculation is estimated to be about 400,000 CPU-hours. These results suggest that, with continued development, it will be feasible for fault-tolerant quantum computers to accelerate the discovery of homogeneous catalysts.

Recent progress in quantum computing is paving the way for the realization of early fault-tolerant quantum computers. To maximize the utility of these devices, it is important to develop quantum algorithms that match their capabilities and limitations. Motivated by this, recent work has developed low-depth quantum algorithms for ground state energy estimation (GSEE), an important subroutine in quantum chemistry and materials. We detail a new GSEE algorithm which, like recent work, uses a number of operations scaling as $O(1/\Delta)$ as opposed to the typical $O(1/\epsilon)$, at the cost of an increase in the number of circuit repetitions from $O(1)$ to $O(1/\epsilon^2)$. The relevant features of this algorithm come about from using a Gaussian window, which exponentially reduces contamination from excited states over the simplest GSEE algorithm based on the Quantum Fourier Transform (QFT). We adapt this algorithm to interpolate between the low-depth and full-depth regime by replacing $\Delta$ with anything between $\Delta$ and $\epsilon$. At the cost of increasing the number of ancilla qubits from $1$ to $O(\log\Delta)$, our method reduces the upper bound on the number of circuit repetitions by a factor of four compared to previous methods.

A milestone in the field of quantum computing will be solving problems in quantum chemistry and materials faster than state-of-the-art classical methods. The current understanding is that achieving quantum advantage in this area will require some degree of fault tolerance. While hardware is improving towards this milestone, optimizing quantum algorithms also brings it closer to the present. Existing methods for ground state energy estimation are costly in that they require a number of gates per circuit that grows exponentially with the desired number of bits in precision. We reduce this cost exponentially, by developing a ground state energy estimation algorithm for which this cost grows linearly in the number of bits of precision. Relative to recent resource estimates of ground state energy estimation for the industrially-relevant molecules of ethylene-carbonate and PF$_6^-$, the estimated gate count and circuit depth is reduced by a factor of 43 and 78, respectively. Furthermore, the algorithm can use additional circuit depth to reduce the total runtime. These features make our algorithm a promising candidate for realizing quantum advantage in the era of early fault-tolerant quantum computing.

Determining the properties of molecules and materials is one of the premier applications of quantum computing. A major question in the field is: how might we use imperfect near-term quantum computers to solve problems of practical value? We propose a quantum algorithm to estimate the properties of molecules using near-term quantum devices. The method is a recursive variational series estimation method, where we expand an operator of interest in terms of Chebyshev polynomials and evaluate each term in the expansion using a variational quantum algorithm. We test our method by computing the one-particle Green's function in the energy domain and the autocorrelation function in the time domain.

Quantum computing is believed to be particularly useful for the simulation of chemistry and materials, among the various applications. In recent years, there have been significant advancements in the development of near-term quantum algorithms for quantum simulation, including VQE and many of its variants. However, for such algorithms to be useful, they need to overcome several critical barriers including the inability to prepare high-quality approximations of the ground state. Current challenges to state preparation, including barren plateaus and the high-dimensionality of the optimization landscape, make state preparation through ansatz optimization unreliable. In this work, we introduce the method of ground state boosting, which uses a limited-depth quantum circuit to reliably increase the overlap with the ground state. This circuit, which we call a booster, can be used to augment an ansatz from VQE or be used as a stand-alone state preparation method. The booster converts circuit depth into ground state overlap in a controllable manner. We numerically demonstrate the capabilities of boosters by simulating the performance of a particular type of booster, namely the Gaussian booster, for preparing the ground state of N2 molecular system. Beyond ground state preparation as a direct objective, many quantum algorithms, such as quantum phase estimation, rely on high-quality state preparation as a subroutine. Therefore, we foresee ground state boosting and similar methods as becoming essential algorithmic components as the field transitions into using early fault-tolerant quantum computers.

We present mathematical and conceptual foundations for the task of robust amplitude estimation using engineered likelihood functions (ELFs), a framework introduced by Wang et al. [PRX Quantum 2, 010346 (2021)] that uses Bayesian inference to enhance the rate of information gain in quantum sampling. These ELFs, which are obtained by choosing tunable parameters in a parametrized quantum circuit to minimize the expected posterior variance of an estimated parameter, play an important role in estimating the expectation values of quantum observables. We give a thorough characterization and analysis of likelihood functions arising from certain classes of quantum circuits and combine this with the tools of Bayesian inference to give a procedure for picking optimal ELF tunable parameters. Finally, we present numerical results to demonstrate the performance of ELFs.

In recent years substantial research effort has been devoted to quantum algorithms for ground state energy estimation (GSEE) in chemistry and materials. Given the many heuristic and non-heuristic methods being developed, it is challenging to assess what combination of these will ultimately be used in practice. One important metric for assessing utility is runtime. For most GSEE algorithms, the runtime depends on the ground state preparation (GSP) method. Towards assessing the utility of various combinations of GSEE and GSP methods, we asked under which conditions a GSP method should be accepted over a reference method, such as the Hartree-Fock state. We introduce a criteria for accepting or rejecting a GSP method for the purposes of GSEE. We consider different GSP methods ranging from heuristics to algorithms with provable performance guarantees and perform numerical simulations to benchmark their performance on different chemical systems, starting from small molecules like the hydrogen atom to larger systems like the jellium. In the future this approach may be used to abandon certain VQE ansatzes and other heursitics. Yet so far our findings do not provide evidence against the use of VQE and more expensive heuristic methods, like the low-depth booster. This work sets a foundation from which to further explore the requirements to achieve quantum advantage in quantum chemistry.

Measuring quantum observables by grouping terms that can be rotated to sums of only products of Pauli ẑ operators (Ising form) is proven to be efficient in near term quantum computing algorithms. This approach requires extra unitary transformations to rotate the state of interest so that the measurement of a fragment's Ising form would be equivalent to the measurement of the fragment for the unrotated state. These extra rotations allow one to perform a fewer number of measurements by grouping more terms into the measurable fragments with a lower overall estimator variance. However, previous estimations of the number of measurements did not take into account nonunit fidelity of quantum gates implementing the additional transformations. Through a circuit fidelity reduction, additional transformations introduce extra uncertainty and increase the needed number of measurements. Here we consider a simple model for errors introduced by additional gates needed in schemes involving groupings of commuting Pauli products. For a set of molecular electronic Hamiltonians, we confirm that the numbers of measurements in schemes using nonlocal qubit rotations are still lower than those in their local qubit rotation counterparts, even after accounting for uncertainties introduced by additional gates.

Quantum chemistry and materials is one of the most promising applications of quantum computing. Yet much work is still to be done in matching industry-relevant problems in these areas with quantum algorithms that can solve them. Most previous efforts have carried out resource estimations for quantum algorithms run on large-scale fault-tolerant architectures, which include the quantum phase estimation algorithm. In contrast, few have assessed the performance of near-term quantum algorithms, which include the variational quantum eigensolver (VQE) algorithm. Recently, a large-scale benchmark study [Gonthier et al. 2020] found evidence that the performance of the variational quantum eigensolver for a set of industry-relevant molecules may be too inefficient to be of practical use. This motivates the need for developing and assessing methods that improve the efficiency of VQE. In this work, we predict the runtime of the energy estimation subroutine of VQE when using robust amplitude estimation (RAE) to estimate Pauli expectation values. Under conservative assumptions, our resource estimation predicts that RAE can reduce the runtime over the standard estimation method in VQE by one to two orders of magnitude. Despite this improvement, we find that the runtimes are still too large to be practical. These findings motivate two complementary efforts towards quantum advantage: 1) the investigation of more efficient near-term methods for ground state energy estimation and 2) the development of problem instances that are of industrial value and classically challenging, but better suited to quantum computation.

Peter D. Johnson,1 Alexander A. Kunitsa,1 Jérôme F. Gonthier,1 Maxwell D. Radin,1 Corneliu Buda,2 Eric J. Doskocil,2 Clena M. Abuan,3 and Jhonathan Romero1 1Zapata Computing, Inc., 100 Federal St., Boston, MA 02110, USA 2Applied Chemistry and Physics Centre of Expertise, BP Group Research, 150 West Warrenville Road, Naperville, IL 60563, USA 3Digital Science and Engineering, BP Innovation and Engineering, 501 Westlake Park Blvd, Houston, TX 77079, USA (Dated: March 15, 2022)

The ability of near-term quantum computers to represent classically-intractable quantum states has brought much interest in using such devices for estimating the ground and excited state energies of fermionic Hamiltonians. The usefulness of such near-term techniques, generally based on the Variational Quantum Eigensolver (VQE), however, is limited by device noise and the need to perform many circuit repetitions. This paper addresses these challenges by generalizing VQE to consider wavefunctions in a subspace spanned by classically tractable states and states that can be prepared on a quantum computer. The manuscript shows how the ground and excited state energies can be estimated using such "classical-boosting" and how this approach can be combined with VQE Hamiltonian decomposition techniques. Unlike existing VQE approaches, the sensitivity to sampling error and device noise approaches zero in the limit where the classically tractable states are able to describe an eigenstate. A detailed analysis of the measurement requirements in the simplest case, where a single computational basis state is used to boost conventional VQE, shows that the ground-state energy estimation of several closed-shell homonuclear diatomic molecules can be accelerated by a factor of approximately 10-1000. The analysis also shows that the measurement reduction of such single basis state boosting, relative to conventional VQE, can be estimated using only the overlap between the ground state and the computational basis state used for boosting.

We formalize the problem of dissipative quantum encoding, and explore the advantages of using Markovian evolution to prepare a quantum code in the desired logical space, with emphasis on discrete-time dynamics and the possibility of exact finite-time convergence. In particular, we investigate robustness of the encoding dynamics and their ability to tolerate initialization errors, thanks to the existence of non-trivial basins of attraction. As a key application, we show that for stabilizer quantum codes on qubits, a finite-time dissipative encoder may always be constructed, by using at most a number of quantum maps determined by the number of stabilizer generators. We find that even in situations where the target code lacks gauge degrees of freedom in its subsystem form, dissipative encoders afford nontrivial robustness against initialization errors, thus overcoming a limitation of purely unitary encoding procedures. Our general results are illustrated in a number of relevant examples, including Kitaev's toric code.

The number of measurements demanded by hybrid quantum-classical algorithms such as the variational quantum eigensolver (VQE) is prohibitively high for many problems of practical value. For such problems, realizing quantum advantage will require methods which dramatically reduce this cost. Previous quantum algorithms that reduce the measurement cost (e.g. quantum amplitude and phase estimation) require error rates that are too low for near-term implementation. Here we propose methods that take advantage of the available quantum coherence to maximally enhance the power of sampling on noisy quantum devices, reducing measurement number and runtime compared to the standard sampling method of the variational quantum eigensolver (VQE). Our scheme derives inspiration from quantum metrology, phase estimation, and the more recent "alpha-VQE" proposal, arriving at a general formulation that is robust to error and does not require ancilla qubits. The central object of this method is what we call the "engineered likelihood function" (ELF), used for carrying out Bayesian inference. We show how the ELF formalism enhances the rate of information gain in sampling as the physical hardware transitions from the regime of noisy intermediate-scale quantum computers into that of quantum error corrected ones. This technique speeds up a central component of many quantum algorithms, with applications including chemistry, materials, finance, and beyond. Similar to VQE, we expect small-scale implementations to be realizable on today's quantum devices.