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Recently, there have been several advancements in quantum algorithms for Gibbs sampling. These algorithms simulate the dynamics generated by an artificial Lindbladian, which is meticulously constructed to obey a detailed-balance condition with the Gibbs state of interest, ensuring it is a stationary point of the evolution, while simultaneously having efficiently implementable time steps. The overall complexity then depends primarily on the mixing time of the Lindbladian, which can vary drastically, but which has been previously bounded in the regime of high enough temperatures [Rouz\'e {\it et al.}~arXiv:2403.12691 and arXiv:2411.04885]. In this work, we calculate the spectral gap of the Lindbladian for free fermions using third quantisation, and then prove a constant gap of the perturbed Lindbladian corresponding to interacting fermions up to some maximal coupling strength. This is achieved by using theorems about stability of the gap for lattice fermions. Our methods apply at any constant temperature and independently of the system size. The gap then provides an upper bound on the mixing time, and hence on the overall complexity of the quantum algorithm, proving that the purified Gibbs state of weakly interacting (quasi-)local fermionic systems of any dimension can be prepared in $\widetilde{\mathcal{O}} (n^3 \operatorname{polylog}(1/\epsilon))$ time on $\mathcal{O}(n)$ qubits, where $n$ denotes the size of the system and $\epsilon$ the desired accuracy. We provide exact numerical simulations for small system sizes supporting the theory and also identify different suitable jump operators and filter functions for the sought-after regime of intermediate coupling in the Fermi-Hubbard model.

We explore the important task of applying a phase $exp(i f(x))$ to a computational basis state $\left| x \right>$. The closely related task of rotating a target qubit by an angle depending on $f(x)$ is also studied. Such operations are key in many quantum subroutines, and often the function $f$ can be well-approximated by a piecewise linear composition. Examples range from the application of diagonal Hamiltonian terms (such as the Coulomb interaction) in grid-based many-body simulation, to derivative pricing algorithms. Here we exploit a parallelisation of the piecewise approach so that all constituent elementary rotations are performed simultaneously, that is, we achieve a total rotation depth of one. Moreover, we explore the use of recursive catalyst 'towers' to implement these elementary rotations efficiently. Depending on the choice of implementation strategy, we find a depth as low as $O(log n + log S)$ for a register of $n$ qubits and a piecewise approximation of $S$ sections. In the limit of multiple repetitions of the oracle, we find that catalyst tower approaches have an $O(S \cdot n)$ T-count, whereas linear interpolation with QROM has an $O(n^{log_2(3)})$ T-count., Comment: 14 pages, references added, table I updated, code for table I added

We present shadow spectroscopy as a simulator-agnostic quantum algorithm for estimating energy gaps using very few circuit repetitions (shots) and no extra resources (ancilla qubits) beyond performing time evolution and measurements. The approach builds on the fundamental feature that every observable property of a quantum system must evolve according to the same harmonic components: we can reveal them by post-processing classical shadows of time-evolved quantum states to extract a large number of time-periodic signals $N_o\propto 10^8$, whose frequencies correspond to Hamiltonian energy differences with Heisenberg-limited precision. We provide strong analytical guarantees that (a) quantum resources scale as $O(\log N_o)$, while the classical computational complexity is linear $O(N_o)$, (b) the signal-to-noise ratio increases with the number of processed signals as $\propto \sqrt{N_o}$, and (c) spectral peak positions are immune to reasonable levels of noise. We demonstrate our approach on model spin systems and the excited state conical intersection of molecular CH$_2$ and verify that our method is indeed intuitively easy to use in practice, robust against gate noise, amiable to a new type of algorithmic-error mitigation technique, and uses orders of magnitude fewer number of shots than typical near-term quantum algorithms -- as low as 10 shots per timestep is sufficient. Finally, we measured a high-quality, experimental shadow spectrum of a spin chain on readily-available IBM quantum computers, achieving the same precision as in noise-free simulations without using any advanced error mitigation, and verified scalability in tensor-network simulations of up to 100-qubit systems., Comment: 33 pages, 14 figures, new results with hardware and figures

The preparation of Hamiltonian eigenstates is essential for many applications in quantum computing; the efficiency with which this can be done is of key interest. A canonical approach exploits the quantum phase estimation (QPE) algorithm. We adopt ideas from variants of this method to implement a resource-frugal iterative scheme, and provide analytic bounds on the complexity (simulation time cost) for various cases of available information and tools. We propose and characterise an extension involving a modification of the target Hamiltonian to increase overall efficiency. The presented methods and bounds are then demonstrated by preparing the ground state of the Hamiltonians of LiH and H$_2$ in second quantisation; we report the performance of both ideal and noisy implementations using simulated quantum computers. Convergence is generally achieved much faster than the bounds suggest, while the qualitative features are validated., Comment: 9 pages, 5 figures

Variational methods offer a highly promising route to exploiting quantum computers for chemistry tasks. Here we employ methods described in a sister paper to the present report, entitled ab initio machine synthesis of quantum circuits, in order to solve problems using adaptively evolving quantum circuits. Consistent with prior authors we find that this approach can outperform human-designed circuits such as the coupled-cluster or hardware-efficient ans\"atze, and we make comparisons for larger instances up to 14 qubits. Moreover we introduce a novel approach to constraining the circuit evolution in the physically relevant subspace, finding that this greatly improves performance and compactness of the circuits. We consider both static and dynamics properties of molecular systems. The emulation environments used is QuESTlink; all resources are open source and linked from this paper., Comment: 8 pages, 8 figures, 8 pages of appendix, and 12 figures in the appendix

Gate-level quantum circuits are often derived manually from higher level algorithms. While this suffices for small implementations and demonstrations, ultimately automatic circuit design will be required to realise complex algorithms using hardware-specific operations and connectivity. Here we explore methods for the ab initio creation of circuits within a machine, either a classical computer or a hybrid quantum-classical device. We consider a range of techniques including: methods for introducing new gate structures, optimisation of parameterised circuits and choices of cost functions, and efficient removal of low-value gates exploiting the quantum geometric tensor and other heuristics. Using these principles we tackle the tasks of automatic encoding of unitary processes and translation (recompilation) of a circuit from one form to another. Using emulated quantum computers with various noise-free gate sets we provide simple examples involving up to 10 qubits, corresponding to 20 qubits in the augmented space we use. Further applications of specific relevance to chemistry modelling are considered in a sister paper, 'Exploiting subspace constraints and ab initio variational methods for quantum chemistry'., Comment: 17 pages, 10 figures

First quantized, grid-based methods for chemistry modelling are a natural and elegant fit for quantum computers. However, it is infeasible to use today's quantum prototypes to explore the power of this approach, because it requires a significant number of near-perfect qubits. Here we employ exactly-emulated quantum computers with up to 36 qubits, to execute deep yet resource-frugal algorithms that model 2D and 3D atoms with single and paired particles. A range of tasks is explored, from ground state preparation and energy estimation to the dynamics of scattering and ionisation; we evaluate various methods within the split-operator QFT (SO-QFT) Hamiltonian simulation paradigm, including protocols previously-described in theoretical papers as well as our own novel techniques. While we identify certain restrictions and caveats, generally the grid-based method is found to perform very well; our results are consistent with the view that first quantized paradigms will be dominant from the early fault-tolerant quantum computing era onward., Comment: Update to reflect published version. 37 pages, 12 figures

A highly anticipated use of quantum computers is the simulation of complex quantum systems including molecules and other many-body systems. One promising method involves directly applying a linear combination of unitaries (LCU) to approximate a Taylor series by truncating after some order. Here we present an adaptation of that method, optimized for Hamiltonians with terms of widely varying magnitude, as is commonly the case in electronic structure calculations. We show that it is more efficient to apply LCU using a truncation that retains larger magnitude terms as determined by an iterative procedure. We obtain bounds on the simulation error for this generalized truncated Taylor method, and for a range of molecular simulations, we report these bounds as well as exact numerical results. We find that our adaptive method can typically improve the simulation accuracy by an order of magnitude, for a given circuit depth., Comment: 14 pages, 7 figures