arXiv subjects

Nathan Fitzpatrick

Publications and source records attributed to Nathan Fitzpatrick.

At least 19 recordsLinked to original sources

arXiv2025

Shorter width truncated Taylor series for Hamiltonian dynamics simulations

As established in the seminal work by Berry et al.[1], expanding the time evolution operator using truncated Taylor series (up to some order $K$) makes a good candidate for simulating Hamiltonian dynamics. Here, we adapt the method but present an alternative quantum circuit that maintains an equivalent asymptotic elementary gate cost but has an exponentially reduced number of ancilla qubits. This is realized by utilizing mid-circuit measurements (with early abort-and-restart of circuit execution), and transforming a series of multi-controlled$(H^k)$ to a series of singly-controlled$(H^{k'})$, where $H$ is a linear combination of unitaries and $k, k'$ are integers. The proposed circuit utilizes a total of $\lceil \log(K) \rceil + \lceil \log(L) \rceil +n$ qubits, where $L$ is the number of terms in the Hamiltonian and $n$ is the system qubit size. Our shorter width circuit with mid-measurements protocol is implemented and evaluated using the programming language Guppy[2,3].

quant-ph↗

arXiv2025

Beyond asymptotic reasoning: the practicalities of a quantum ground state projector based on the wall-Chebyshev expansion

We consider a quantum algorithm for ground-state preparation based on a Chebyshev series approximation to the wall function. In a classical setting, this approach is appealing as it guarantees rapid convergence. We analyze the asymptotic scaling and success probabilities of different quantum implementations and provide numerical benchmarks, comparing the performance of the wall-Chebyshev projectors with current state-of-the-art approaches. We find that this approach requires fewer serial applications of the Hamiltonian oracle to achieve a given ground state fidelity, but is severely limited by exponentially decaying success probability. However, we find that some implementations maintain non-trivial success probability in regimes where wall-Chebyshev projection leads to a fidelity improvement over other approaches. As the wall-Chebyshev projector is highly robust to loose known upper bounds on the true ground state energy, it offers a potential resource trade-off, particulary in the early fault-tolerant regime of quantum computation.

quant-ph↗

arXiv2025

The Quantum Paldus Transform: Efficient Circuits with Applications

We present the Quantum Paldus Transform: an efficient quantum algorithm for block-diagonalising fermionic, spin-free Hamiltonians in the second quantisation. Our algorithm implements an isometry between the occupation number basis of a fermionic Fock space of $2d$ modes, and the Gelfand-Tsetlin (GT) states spanning irreducible representations of the group $U(d) \times SU(2)$. The latter forms a basis indexed by well-defined values of total particle number $N$, global spin $S$, spin projection $M$, and $U(d)$ GT patterns. This realises the antisymmetric unitary-unitary duality discovered by Howe and developed into the Unitary Group Approach (UGA) for computational chemistry by Paldus and Shavitt in the 1970s. The Paldus transform lends tools from the UGA readily applicable to quantum computational chemistry, leading to maximally sparse representations of spin-free Hamiltonians, efficient preparation of Configuration State Functions, and a direct interpretation of quantum chemistry reduced density matrix elements in terms of $SU(2)$ angular momentum coupling. The transform also enables the encoding of quantum information into novel Decoherence-Free Subsystems for use in communication and error mitigation. Our work can be seen as a generalisation of the quantum Schur transform for the second quantisation, made tractable by the Pauli exclusion principle. Alongside self-contained derivations of the underlying dualities we provide fault-tolerant circuit compilation methods with full gate counts for the Paldus transform, resulting in $\mathcal{O}(d^3)$ Toffoli complexity, where a transform on $50$ spatial orbitals would require a modest $5500$ Toffoli gates. This paves the way for significant advancements in quantum simulation on quantum computers enabled by the UGA paradigm.

quant-ph↗

arXiv2025

Hamiltonian dynamics simulation using linear combination of unitaries on an ion trap quantum computer

The linear combination of unitaries (LCU) method has proven to scale better than existing product formulas in simulating long time Hamiltonian dynamics. However, given the number of multi-control gate operations in the standard prepare-select-unprepare architecture of LCU, it is still resource-intensive to implement on the current quantum computers. In this work, we demonstrate LCU implementations on an ion trap quantum computer for calculating squared overlaps $|\langle \psi(t=0)|\psi(t>0)\rangle|^2$ of time-evolved states. This is achieved by an optimized LCU method, based on pre-selecting relevant unitaries, coupled with a compilation strategy which makes use of quantum multiplexor gates, leading to a significant reduction in the depth and number of two-qubit gates in circuits. For $L$ Pauli strings in a Taylor series expanded $n$-qubit-mapped time evolution operator, we find a two-qubit gate count of $2^{\lceil log_2(L)\rceil}(2n+1)-n-2$. We test this approach by simulating a Rabi-Hubbard Hamiltonian.

quant-ph↗

arXiv2024

Realizing Quantum Kernel Models at Scale with Matrix Product State Simulation

Data representation in quantum state space offers an alternative function space for machine learning tasks. However, benchmarking these algorithms at a practical scale has been limited by ineffective simulation methods. We develop a quantum kernel framework using a Matrix Product State (MPS) simulator and employ it to perform a classification task with 165 features and 6400 training data points, well beyond the scale of any prior work. We make use of a circuit ansatz on a linear chain of qubits with increasing interaction distance between qubits. We assess the MPS simulator performance on CPUs and GPUs and, by systematically increasing the qubit interaction distance, we identify a crossover point beyond which the GPU implementation runs faster. We show that quantum kernel model performance improves as the feature dimension and training data increases, which is the first evidence of quantum model performance at scale.

quant-ph↗

arXiv2024

Unification of Finite Symmetries in Simulation of Many-body Systems on Quantum Computers

Symmetry is fundamental in the description and simulation of quantum systems. Leveraging symmetries in classical simulations of many-body quantum systems can results in significant overhead due to the exponentially growing size of some symmetry groups as the number of particles increases. Quantum computers hold the promise of achieving exponential speedup in simulating quantum many-body systems; however, a general method for utilizing symmetries in quantum simulations has not yet been established. In this work, we present a unified framework for incorporating symmetry group transforms on quantum computers to simulate many-body systems. The core of our approach lies in the development of efficient quantum circuits for symmetry-adapted projection onto irreducible representations of a group or pairs of commuting groups. We provide resource estimations for common groups, including the cyclic and permutation groups. Our algorithms demonstrate the capability to prepare coherent superpositions of symmetry-adapted states and to perform quantum evolution across a wide range of models in condensed matter physics and \textit{ab initio} electronic structure in quantum chemistry. Specifically, we execute a symmetry-adapted quantum subroutine for small molecules in first-quantization on noisy hardware. In addition, we present a discussion of open problems regarding treating symmetries in digital quantum simulations of many-body systems, paving the way for future systematic investigations into leveraging symmetries \emph{quantumly} for practical quantum advantage. The broad applicability and rigorous resource estimation for symmetry transformations make our framework appealing for achieving provable quantum advantage on fault-tolerant quantum computers, especially for symmetry-related properties.

quant-ph↗

arXiv2024

Recovering complex ecological dynamics from time series using state-space universal dynamic equations

Ecological systems often exhibit complex nonlinear dynamics like oscillations, chaos, and regime shifts. Universal dynamic equations have shown promise in modeling complex dynamics by combining known functional forms with neural networks that represent unknown relationships. However, these methods do not yet accommodate the forms of uncertainty common to ecological datasets. To address this limitation, we developed state-space universal dynamic equations by combining universal differential equations with a state-space modeling framework, accounting for uncertainty. We tested this framework on two simulated and two empirical case studies and found that this method can recover nonlinear biological interactions that produce complex behaviors, including chaos and regime shifts. Their forecasting performance is context-dependent, with the best performance being achieved on chaotic and oscillating time series. This new approach leveraging both ecological theory and data-driven machine learning offers a promising new way to make accurate and useful predictions of ecosystem change.

q-bio.PE↗

arXiv2024

Quantum state preparation for multivariate functions

A fundamental step of any quantum algorithm is the preparation of qubit registers in a suitable initial state. Often qubit registers represent a discretization of continuous variables and the initial state is defined by a multivariate function. We develop protocols for preparing quantum states whose amplitudes encode multivariate functions by linearly combining block-encodings of Fourier and Chebyshev basis functions. Without relying on arithmetic circuits, quantum Fourier transforms, or multivariate quantum signal processing, our algorithms are simpler and more effective than previous proposals. We analyze requirements both asymptotically and pragmatically in terms of near/medium-term resources. Numerically, we prepare bivariate Student's t-distributions, 2D Ricker wavelets and electron wavefunctions in a 3D Coulomb potential, which are initial states with potential applications in finance, physics and chemistry simulations. Finally, we prepare bivariate Gaussian distributions on the Quantinuum H2-1 trapped-ion quantum processor using 24 qubits and up to 237 two-qubit gates.

quant-ph↗

arXiv2024

Quantum Circuit Optimization with AlphaTensor

A key challenge in realizing fault-tolerant quantum computers is circuit optimization. Focusing on the most expensive gates in fault-tolerant quantum computation (namely, the T gates), we address the problem of T-count optimization, i.e., minimizing the number of T gates that are needed to implement a given circuit. To achieve this, we develop AlphaTensor-Quantum, a method based on deep reinforcement learning that exploits the relationship between optimizing T-count and tensor decomposition. Unlike existing methods for T-count optimization, AlphaTensor-Quantum can incorporate domain-specific knowledge about quantum computation and leverage gadgets, which significantly reduces the T-count of the optimized circuits. AlphaTensor-Quantum outperforms the existing methods for T-count optimization on a set of arithmetic benchmarks (even when compared without making use of gadgets). Remarkably, it discovers an efficient algorithm akin to Karatsuba's method for multiplication in finite fields. AlphaTensor-Quantum also finds the best human-designed solutions for relevant arithmetic computations used in Shor's algorithm and for quantum chemistry simulation, thus demonstrating it can save hundreds of hours of research by optimizing relevant quantum circuits in a fully automated way.

quant-ph↗

arXiv2023

Quantum Computed Green's Functions using a Cumulant Expansion of the Lanczos Method

In this paper, we present a quantum computational method to calculate the many-body Green's function matrix in a spin orbital basis. We apply our approach to finite-sized fermionic Hubbard models and related impurity models within Dynamical Mean Field Theory, and demonstrate the calculation of Green's functions on Quantinuum's H1-1 trapped-ion quantum computer. Our approach involves a cumulant expansion of the Lanczos method, using Hamiltonian moments as measurable expectation values. This bypasses the need for a large overhead in the number of measurements due to repeated applications of the variational quantum eigensolver (VQE), and instead measures the expectation value of the moments with one set of measurement circuits. From the measured moments, the tridiagonalised Hamiltonian matrix can be computed, which in turn yields the Green's function via continued fractions. While we use a variational algorithm to prepare the ground state in this work, we note that the modularity of our implementation allows for other (non-variational) approaches to be used for the ground state.

cond-mat.str-el↗

arXiv2023

Calculating the Single-Particle Many-body Green's Functions via the Quantum Singular Value Transform Algorithm

The Quantum Singular Value Transformation (QSVT) is a technique that provides a unified framework for describing many of the quantum algorithms discovered to date. We implement a noise-free simulation of the technique to investigate how it can be used to perform matrix inversion, which is an important step in calculating the single-particle Green's function in the Lehmann representation. Due to the inverse function not being defined at zero, we explore the effect of approximating f(x)=1/x with a polynomial. This is carried out by calculating the single-particle Green's function of the two-site single-impurity Anderson model. We also propose a new circuit construction for the linear combination of unitaries block encoding technique, that reduces the number of single and two-qubit gates required.

quant-ph↗

arXiv2023

Non-unitary Trotter circuits for imaginary time evolution

We propose an imaginary time equivalent of the well-established Pauli gadget primitive for Trotter-decomposed real time evolution, using mid-circuit measurements on a single ancilla qubit. Imaginary time evolution (ITE) is widely used for obtaining the ground state of a system on classical hardware, computing thermal averages, and as a component of quantum algorithms that perform non-unitary evolution. Near-term implementations on quantum hardware rely on heuristics, compromising their accuracy. As a result, there is growing interest in the development of more natively quantum algorithms. Since it is not possible to implement a non-unitary gate deterministically, we resort to the implementation of probabilistic imaginary time evolution (PITE) algorithms, which rely on a unitary quantum circuit to simulate a block encoding of the ITE operator - that is, they rely on successful ancillary measurements to evolve the system non-unitarily. Compared with previous PITE proposals, the suggested block encoding in this paper results in shorter circuits and is simpler to implement, requiring only a slight modification of the Pauli gadget primitive. This scheme was tested on the transverse Ising model and the fermionic Hubbard model and is demonstrated to converge to the ground state of the system.

quant-ph↗

arXiv2023

Simulating non-unitary dynamics using quantum signal processing with unitary block encoding

We adapt a recent advance in resource-frugal quantum signal processing - the Quantum Eigenvalue Transform with Unitary matrices (QET-U) - to explore non-unitary imaginary time evolution on early fault-tolerant quantum computers using exactly emulated quantum circuits. We test strategies for optimising the circuit depth and the probability of successfully preparing the desired imaginary-time evolved states. For the task of ground state preparation, we confirm that the probability of successful post-selection is quadratic in the initial reference state overlap $\gamma$ as $O(\gamma^2)$. When applied instead to thermal state preparation, we show QET-U can directly estimate partition functions at exponential cost. Finally, we combine QET-U with Trotter product formula to perform non-normal Hamiltonian simulation in the propagation of Lindbladian open quantum system dynamics. We find that QET-U for non-unitary dynamics is flexible, intuitive and straightforward to use, and suggest ways for delivering quantum advantage in simulation tasks.

quant-ph↗

arXiv2022

Variational Phase Estimation with Variational Fast Forwarding

Subspace diagonalisation methods have appeared recently as promising means to access the ground state and some excited states of molecular Hamiltonians by classically diagonalising small matrices, whose elements can be efficiently obtained by a quantum computer. The recently proposed Variational Quantum Phase Estimation (VQPE) algorithm uses a basis of real time-evolved states, for which the energy eigenvalues can be obtained directly from the unitary matrix U = exp(-iHt), which can be computed with cost linear in the number of states used. In this paper, we report a circuit-based implementation of VQPE for arbitrary molecular systems and assess its performance and costs for the H2, H3+ and H6 molecules. We also propose using Variational Fast Forwarding (VFF) to decrease to quantum depth of time-evolution circuits for use in VQPE. We show that the approximation provides a good basis for Hamiltonian diagonalisation even when its fidelity to the true time evolved states is low. In the high fidelity case, we show that the approximate unitary U can be diagonalised instead, preserving the linear cost of exact VQPE.

quant-ph↗

arXiv2021

A case study of variational quantum algorithms for a job shop scheduling problem

Combinatorial optimization models a vast range of industrial processes aiming at improving their efficiency. In general, solving this type of problem exactly is computationally intractable. Therefore, practitioners rely on heuristic solution approaches. Variational quantum algorithms are optimization heuristics that can be demonstrated with available quantum hardware. In this case study, we apply four variational quantum heuristics running on IBM's superconducting quantum processors to the job shop scheduling problem. Our problem optimizes a steel manufacturing process. A comparison on 5 qubits shows that the recent filtering variational quantum eigensolver (F-VQE) converges faster and samples the global optimum more frequently than the quantum approximate optimization algorithm (QAOA), the standard variational quantum eigensolver (VQE), and variational quantum imaginary time evolution (VarQITE). Furthermore, F-VQE readily solves problem sizes of up to 23 qubits on hardware without error mitigation post processing.

quant-ph↗

arXiv2021

Reducing Unitary Coupled Cluster Circuit Depth by Classical Stochastic Amplitude Pre-Screening

Unitary Coupled Cluster (UCC) approaches are an appealing route to utilising quantum hardware to perform quantum chemistry calculations, as quantum computers can in principle perform UCC calculations in a polynomially scaling fashion, as compared to the exponential scaling required on classical computers. Current noisy intermediate scale quantum (NISQ) computers are limited by both hardware capacity in number of logical qubits and the noise introduced by the deep circuits required for UCC calculations using the Variational Quantum Eigensolver (VQE) approach. We present a combined classical--quantum approach where a stochastic classical UCC pre-processing step is used to determine the important excitations in the UCC ansatz. The reduced number of selected excitations are then used in a UCC-based VQE calculation. This approach gives a systematically improvable approximation, and we show that significant reductions in quantum resources can be achieved, with simulations on the CH$_2$, N$_2$ and N$_2$H$_2$ molecules giving sub-milliHartree errors.

quant-ph↗

arXiv2021

Evaluating low-depth quantum algorithms for time evolution on fermion-boson systems

Simulating time evolution of quantum systems is one of the most promising applications of quantum computing and also appears as a subroutine in many applications such as Green's function methods. In the current era of NISQ machines we assess the state of algorithms for simulating time dynamics with limited resources. We propose the Jaynes-Cummings model and extensions to it as useful toy models to investigate time evolution algorithms on near-term quantum computers. Using these simple models, direct Trotterisation of the time evolution operator produces deep circuits, requiring coherence times out of reach on current NISQ hardware. Therefore we test two alternative responses to this problem: variational compilation of the time evolution operator, and variational quantum simulation of the wavefunction ansatz. We demonstrate numerically to what extent these methods are successful in time evolving this system. The costs in terms of circuit depth and number of measurements are compared quantitatively, along with other drawbacks and advantages of each method. We find that computational requirements for both methods make them suitable for performing time evolution simulations of our models on NISQ hardware. Our results also indicate that variational quantum compilation produces more accurate results than variational quantum simulation, at the cost of a larger number of measurements.

quant-ph↗

arXiv2021

Molecular Excited State Calculations with Adaptive Wavefunctions on a Quantum Eigensolver Emulation: Reducing Circuit Depth and Separating Spin States

Ab initio electronic excited state calculations are necessary for the quantitative study of photochemical reactions, but their accurate computation on classical computers is plagued by prohibitive scaling. The Variational Quantum Deflation (VQD) is an extension of the Variational Quantum Eigensolver (VQE) for calculating electronic excited state energies, and has the potential to address some of these scaling challenges using quantum computers. However, quantum computers available in the near term can only support a limited number of circuit operations, so reducing the quantum computational cost in VQD methods is critical to their realisation. In this work, we investigate the use of adaptive quantum circuit growth (ADAPT-VQE) in excited state VQD calculations, a strategy that has been successful previously in reducing the resource for ground state energy VQE calculations. We also invoke spin restrictions to separate the recovery of eigenstates with different spin symmetry to reduce the number of calculations and accumulation of errors. We created a quantum eigensolver emulation package - Quantum Eigensolver Building on Achievements of Both quantum computing and quantum chemistry (QEBAB) - for testing the proposed adaptive procedure against two VQD methods that use fixed-length quantum circuits. For a lithium hydride test case we found that the spin-restricted adaptive growth variant of VQD uses the most compact circuits by far, consistently recovers adequate electron correlation energy for different nuclear geometries and eigenstates while isolating the singlet and triplet manifold. This work is a further step towards developing techniques which improve the efficiency of hybrid quantum algorithms for excited state quantum chemistry, opening up the possibility of exploiting real quantum computers for electronic excited state calculations sooner than previously anticipated.

quant-ph↗