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arXiv:2504.15841

Quantum Discrete Variable Representations

Szymon Pliś, E. J. Zak

Published Apr 22, 2025• 1 citation
arXivHTMLPDF

Abstract

We present a fault-tolerant quantum algorithm for implementing the Discrete Variable Representation (DVR) transformation, a technique widely used in simulations of quantum-mechanical Hamiltonians. DVR provides a diagonal representation of local operators and enables sparse Hamiltonian structures, making it a powerful alternative to the finite basis representation (FBR), particularly in high-dimensional problems. While DVR has been extensively used in classical simulations, its quantum implementation, particularly using Gaussian quadrature grids, remains underexplored. We develop a quantum circuit that efficiently transforms FBR into DVR by following a recursive construction based on quantum arithmetic operations, and we compare this approach with methods that directly load DVR matrix elements using quantum read-only memory (QROM). We analyze the quantum resources, including T-gate and qubit counts, required for implementing the DVR unitary and discuss preferable choices of QROM-based and recursive-based methods for a given matrix size and precision. This study lays the groundwork for utilizing DVR Hamiltonians in quantum algorithms such as quantum phase estimation with block encoding.

DOI
10.48550/arxiv.2504.15841

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Abstract We present a quantum algorithm for simulating rovibrational Hamiltonians on fault-tolerant quantum computers. The method integrates exact curvilinear kinetic energy operators and general-form potential energy surfaces expressed in a hybrid finite-basis/discrete-variable representation. The Hamiltonian is encoded as a unitary quantum circuit using a quantum read-only memory construction based on the Walsh–Hadamard transform, enabling high-accuracy quantum phase estimation of rovibrational energy levels and dynamics simulations. Our technique provides asymptotic reductions in both logical qubit count and T-gate complexity that are exponential in the number of atoms and at least polynomial in the total Hilbert-space size, relative to existing block-encoding techniques based on linear combinations of unitaries and variational basis representation. Compared with classical variational methods, it offers exponential memory savings and polynomial reductions in time complexity. The quantum volume required for computing the rovibrational spectrum of water can be reduced by up to 10 5 × compared with other quantum methods, increasing to at least 10 6 for a classically intractable 30-dimensional (12-atom) molecular system, with fewer than 300 logical qubits.
Given the prevalence of superconducting platforms for uses in quantum computing and quantum sensing, the simulation of quantum superconducting circuits has become increasingly important for identifying system characteristics and modeling their relevant dynamics. Various numerical tools and software packages have been developed with this purpose in mind, typically utilizing the harmonic oscillator basis or the charge basis to represent a Hamiltonian. In this work, we instead consider the use of discrete variable representations (DVRs) to model superconducting circuits. In particular, we use `sinc DVRs' of both charge number and phase to approximate the eigenenergies of several prototypical examples, exploring their use and effectiveness in the numerical analysis of superconducting circuits. We find that not only are these DVRs capable of achieving decoherence-accurate simulation, i.e., accuracy at the resolution of experiments subject to decay, decoherence, and dephasing, they also demonstrate improvements in efficiency with smaller basis sizes and better convergence over standard approaches, showing that DVRs are an advantageous alternative for representing superconducting circuits.
We present a numerical implementation of the quantum chemistry density matrix renormalization group (DMRG) using the hybrid discrete variable representation (DVR)/Gaussian basis set. The z -axis of real space is discretized by a DVR basis set and each transversal plane is described by the eigenstates of the transversal core Hamiltonian, represented in a set of primitive Gaussian basis functions. Such a hybrid basis can reduce the computation of two-electron repulsion integrals. One main advantage of using the local DVR basis sets over finite-difference methods for real space discretization is that the kinetic energy operator matrix elements can be computed exactly. This method is first applied to a one-dimensional pseudo-hydrogen chain under screened Coulomb potential and then to a realistic hydrogen chain, whereby the DMRG/DVR results are comparable in accuracy to the full configuration interaction results.
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We introduce Quantum Index Algebra (QIA) as a finite, index-based algebraic framework for representing and manipulating quantum operators on Hilbert spaces of dimension 2m2^m2m. In QIA, operators are expressed as structured combinations of basis elements indexed by Boolean codes, allowing products, commutators, and conjugations to be computed through finite rules on discrete indices rather than through dense matrix arithmetic. This representation unifies combinatorial index structure, explicit matrix realization, and transformation properties under Walsh-Hadamard-type transforms within a single formalism. Using QIA and its associated block-matrix realization, we reformulate the Bernstein-Vazirani hidden-string problem in its phase-oracle form entirely within a real, finite-dimensional algebraic setting. We show that, under structured oracle access, the QIA procedure reproduces the Bernstein-Vazirani algorithm exactly and achieves the same asymptotic query complexity and circuit depth as the standard quantum algorithm. In particular, the hidden string is recovered by symbolic manipulation of a sparse algebraic representation of the oracle rather than by numerical simulation of quantum amplitudes. Our results demonstrate that the apparent quantum speed-up in this setting is a consequence of operator structure rather than Hilbert-space dimensionality alone. QIA thus provides a precise language for separating genuinely quantum resources from those arising from algebraic and combinatorial structures and offers a new perspective on the classical simulability of structured quantum circuits.