arXiv subjects

Matthew Pocrnic

Publications and source records attributed to Matthew Pocrnic.

8 recordsLinked to original sources

arXiv2026

Improved constant factors for qubitized Hamiltonian simulation

Quantum signal processing (QSP) serves as the asymptotically optimal technique for Hamiltonian simulation on a quantum computer. By approximating the time evolution operator via the Jacobi-Anger expansion, the Hamiltonian simulation problem reduces to a problem in polynomial approximation theory: find a sufficient degree-$d$ polynomial series to approximate $e^{-i\tau x}$ on $[-1,1]$ within error $\epsilon$. While $d\in\tilde{\mathcal{O}}(\tau)$ is known to be asymptotically optimal, there exists a gap between state-of-the-art bounds and the optimal constant multiplicative factor, which is approximately equal to 1. Here, we close this gap almost entirely, to the point where possible future improvements will not be of practical significance. Our improvement resides in a careful treatment of the Bessel tail in the Jacobi-Anger series using Kapteyn's and Watson's inequalities, thereby reducing the overhead estimates for all Hamiltonian simulation tasks on quantum computers by a factor of $\approx e/2$.

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arXiv2026

Halving the cost of QROM

Table lookup, often referred to as quantum read only memory (QROM), is one of the most widely used subroutines in quantum algorithms, and constitutes the majority share of algorithmic overheads in most practical applications of quantum computers. It involves the coherent loading of $N$ bitstrings of length $b$ in superposition, and naively has a non-Clifford cost of $N$ Toffolis. It is known that given access to $b\, \lambda$ dirty qubits, one can reduce the Toffoli cost of QROM to $2\frac{N}{\lambda} + 4b(\lambda - 1)$. In this work, we first present an optimization to reduce this cost to $2\frac{N}{\lambda} + 2b(\lambda - 1) + 2\lambda-6$ by replacing the ``SelectSwap" architecture with ``SelectCopy". We then provide a further optimization for the qubit-constrained regime where the Toffoli cost is typically $\sim 2\frac{N}{\lambda}$, and reduce it to $\sim (1+\frac{1}{b})\frac{N}{\lambda}$, cutting the cost by approximately $50\%$ and effectively matching the performance of clean-qubit QROM using dirty qubits for practical values of $b$. Lastly, we provide a parametric family of methods that allow the interpolation of the prefactor of the $\frac{N}{\lambda} $ term from $2$ to ($\, 1+\frac{1}{b}\,$) to obtain the best cost for different qubit availability regimes.

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arXiv2026

Efficient Quantum Simulation for Nonlinear Stochastic Differential Equations

Nonlinear stochastic differential equations (NSDEs) are a pillar of mathematical modeling for scientific and engineering applications. Accurate and efficient simulation of large-scale NSDEs is prohibitive on classical computers due to the large number of degrees of freedom, and it is challenging on quantum computers due to the linear and unitary nature of quantum mechanics. We develop a quantum algorithm to tackle nonlinear differential equations driven by the Ornstein-Uhlenbeck (OU) stochastic process. The query complexity of our algorithm scales logarithmically with the error tolerance and nearly quadratically with the simulation time. Our algorithmic framework comprises probabilistic Carleman linearization (PCL) to tackle nonlinearity coupled with stochasticity, and stochastic linear combination of Hamiltonian simulations (SLCHS) to simulate stochastic non-unitary dynamics. We obtain probabilistic exponential convergence for the Carleman linearization of Liu et al. [1], provided the NSDE is stable and reaches a steady state. We extend deterministic LCHS to stochastic linear differential equations, retaining near-optimal parameter scaling from An et al. [2] except for the nearly quadratic time scaling. This is achieved by using Monte Carlo integration for time discretization of both the stochastic inhomogeneous term in LCHS and the truncated Dyson series for each Hamiltonian simulation.

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arXiv2026

Efficient Simulation of Pre-Born-Oppenheimer Dynamics on a Quantum Computer

In this work, we present a quantum algorithm for direct first-principles simulation of electron-nuclear dynamics on a first-quantized real-space grid. Our algorithm achieves best-in-class efficiency for block-encoding the pre-Born-Oppenheimer molecular Hamiltonian by harnessing the linear scaling of swap networks for implementing the quadratic number of particle interactions, while using a novel alternating sign implementation of the Coulomb interaction that exploits highly optimized arithmetic routines. We benchmark our approach for a series of scientifically and industrially relevant chemical reactions. We demonstrate over an order-of-magnitude reduction in costs compared to previous state-of-the-art for the $\rm NH_3+BF_3$ reaction, achieving a Toffoli cost of $8.7\times10^{9}$ per femtosecond using $1362$ logical qubits (system + ancillas). Our results significantly lower the resources required for fault-tolerant simulations of photochemical reactions, while providing a suite of algorithmic primitives that are expected to serve as foundational building blocks for a broader class of quantum algorithms.

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arXiv2025

Discovery of energy landscapes towards optimized quantum transport: Environmental effects and long-range tunneling

Carrier transport in quantum networks is governed by a variety of factors, including network dimensionality and connectivity, on-site energies, couplings between sites and whether they are short- or long-range, leakage processes, and environmental effects. In this work, we identify classes of quasi-one-dimensional chains with energy profiles that optimize carrier transport under such influences. Specifically, we optimize on-site energies using Optax's optimistic gradient descent and AdaMax algorithms, enabled by the JAX automatic differentiation framework. Focusing on nonequilibrium steady-state transport, we study the system's behavior under combined unitary and nonunitary (dephasing and dissipative) effects using the Lindblad quantum master equation. After validating our optimization scheme on short chains, we extend the study to larger systems where we identify systematic patterns in energy profiles. Our analysis reveals that different types of energy landscapes enhance transport, depending on whether inter-site tunneling couplings in the chain are short- or long-range, the existence of environmental interactions, and the temperature of the environment. Our classification and insights of optimal energy landscapes offer guidance for designing efficient transport systems for electronic, photovoltaic and quantum communication applications.

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arXiv2025

Constant-Factor Improvements in Quantum Algorithms for Linear Differential Equations

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 100-200x reduction in runtime costs. This analysis contributes towards establishing more promising applications for quantum computing.

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arXiv2023

Quantum Simulation of Lindbladian Dynamics via Repeated Interactions

The Lindblad equation generalizes the Schr\"{o}dinger equation to quantum systems that undergo dissipative dynamics. The quantum simulation of Lindbladian dynamics is therefore non-unitary, preventing a naive application of state-of-the-art quantum algorithms. Here, we make use of an approximate correspondence between Lindbladian dynamics and evolution based on Repeated Interaction (RI) CPTP maps to write down a Hamiltonian formulation of the Lindblad dynamics and derive a rigorous error bound on the master equation. Specifically, we show that the number of interactions needed to simulate the Liouvillian $e^{t\mathcal{L}}$ within error $\epsilon$ scales in a weak coupling limit as $\nu\in O(t^2\|\mathcal{L}\|_{1\rightarrow 1}^2/\epsilon)$. This is significant because the error in the Lindbladian approximation to the dynamics is not explicitly bounded in existing quantum algorithms for open system simulations. We then provide quantum algorithms to simulate RI maps using an iterative Qubitization approach and Trotter-Suzuki formulas and specifically show that for iterative Qubitization the number of operations needed to simulate the dynamics (for a fixed value of $\nu$) scales in a weak coupling limit as $O(\alpha_0 t + \nu \log(1/\epsilon)/\log\log(1/\epsilon))$ where $\alpha_0$ is the coefficient $1$-norm for the system and bath Hamiltonians. This scaling would appear to be optimal if the complexity of $\nu$ is not considered, which underscores the importance of considering the error in the Liouvillian that we reveal in this work.

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arXiv2023

Composite QDrift-Product Formulas for Quantum and Classical Simulations in Real and Imaginary Time

Recent work has shown that it can be advantageous to implement a composite channel that partitions the Hamiltonian $H$ for a given simulation problem into subsets $A$ and $B$ such that $H=A+B$, where the terms in $A$ are simulated with a Trotter-Suzuki channel and the $B$ terms are randomly sampled via the QDrift algorithm. Here we show that this approach holds in imaginary time, making it a candidate classical algorithm for quantum Monte-Carlo calculations. We upper-bound the induced Schatten-$1 \to 1$ norm on both imaginary-time QDrift and Composite channels. Another recent result demonstrated that simulations of Hamiltonians containing geometrically-local interactions for systems defined on finite lattices can be improved by decomposing $H$ into subsets that contain only terms supported on that subset of the lattice using a Lieb-Robinson argument. Here, we provide a quantum algorithm by unifying this result with the composite approach into ``local composite channels" and we upper bound the diamond distance. We provide exact numerical simulations of algorithmic cost by counting the number of gates of the form $e^{-iH_j t}$ and $e^{-H_j \beta}$ to meet a certain error tolerance $\epsilon$. We show constant factor advantages for a variety of interesting Hamiltonians, the maximum of which is a $\approx 20$ fold speedup that occurs for a simulation of Jellium.

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