arXiv subjects

Dhrumil Patel

Publications and source records attributed to Dhrumil Patel.

At least 19 recordsLinked to original sources

arXiv2026

Quantum thermodynamics and semidefinite optimization: Boltzmann, Fermi-Dirac, and Bose-Einstein frameworks

Here we argue how quantum thermodynamics offers a unifying interpretation for a wide class of semidefinite programs (SDPs) that arise in quantum information. Three SDP variable constraints, namely trace-one density operators, operator-bounded measurements, and the unbounded positive semidefinite cone, admit thermodynamic regularizations associated with Boltzmann, Fermi-Dirac, and Bose-Einstein statistics, respectively. In each case the entropy-regularized dual is unconstrained and concave in a chemical-potential vector, and the primal optimum is a thermal operator of the matched statistics. The dual gradient and Hessian are thermal expectation values, enabling hybrid quantum-classical algorithms for solving a wide variety of SDPs.

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arXiv2026

Improved sample complexity bound for sample-based Lindbladian simulation

We establish improved sample-complexity bounds for sample-based Lindbladian simulation based on the Wave Matrix Lindbladization (WML) algorithm. For a jump operator $L$ with dimension $d$, we derive an explicit non-asymptotic sample complexity bound $n_d^*(t,\varepsilon) \le \left( \frac{2d+3}{8} \right) \|L\|_\infty^2 \left( \frac{t^2}{\varepsilon} \right)$, holding for simulation time $t$ and error $\varepsilon$. This refines the dimension dependence of the best previously known bound, $O(d^2 t^2/\varepsilon)$, from [Go et al., Quantum Sci. Tech. 10, 045058 (2025)]. Remarkably, we show that this dimensional overhead can be entirely avoided when $\| L\|_\infty^2 = O(1/d)$, a condition satisfied with high probability for random Lindblad operators, yielding a typical-case sample complexity of $O(t^2/\varepsilon)$. On the other hand, in the worst case, we show that WML necessarily requires $\Omega(dt^2/\varepsilon)$ samples by constructing an explicit example with a rank-one Lindblad operator. Our results reveal a sharp dichotomy between typical and adversarial sample complexities in Lindbladian simulation, thereby strengthening the theoretical foundations of sample-based quantum algorithms.

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arXiv2026

Quantum Phaselift

Estimating quantum time-series such as the Loschmidt amplitude $f(t)=\langle\psi|\mathrm{e}^{-\mathrm{i}Ht}|\psi\rangle$ is central to spectroscopy, Hamiltonian analysis, and many phase-estimation algorithms. Direct estimation via the Hadamard test requires controlled implementations of $\mathrm{e}^{-\mathrm{i}Ht}$, and the depth of these controlled circuits grows with $t$, making long-time estimation challenging on near-term hardware. We introduce Quantum Phaselift, a lifting-based framework that estimates the rank-one matrix $Z = f f^\dagger$ rather than estimating $f$ directly. We propose simple quantum circuits for estimating the entries of $Z$ and show that measuring only a narrow band of this matrix around the diagonal is sufficient to uniquely recover $f$. Crucially, this reformulation decouples the controlled circuit depth from the maximum evolution time to scale instead with the width of the measured band. We prove that a $O(1)$ bandwidth suffices for generic signals, leading to substantial savings in controlled operations compared to direct estimation methods. We develop three recovery algorithms with provable exact recovery in the noiseless setting and stability under measurement noise. Finally, we numerically demonstrate that high-quality recovery is possible for the 2D Fermi-Hubbard and 2D transverse-field Ising model signals of size exceeding 100 time points using only a few million measurement shots and reasonable post-processing time, making our time-series estimation techniques efficient and effective for near-term implementations.

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arXiv2025

Isolated Sign Language Recognition with Segmentation and Pose Estimation

The recent surge in large language models has automated translations of spoken and written languages. However, these advances remain largely inaccessible to American Sign Language (ASL) users, whose language relies on complex visual cues. Isolated sign language recognition (ISLR) - the task of classifying videos of individual signs - can help bridge this gap but is currently limited by scarce per-sign data, high signer variability, and substantial computational costs. We propose a model for ISLR that reduces computational requirements while maintaining robustness to signer variation. Our approach integrates (i) a pose estimation pipeline to extract hand and face joint coordinates, (ii) a segmentation module that isolates relevant information, and (iii) a ResNet-Transformer backbone to jointly model spatial and temporal dependencies.

cs.CV↗

arXiv2025

Fermionic dynamics on a trapped-ion quantum computer beyond exact classical simulation

Simulation of the time-dynamics of fermionic many-body systems has long been predicted to be one of the key applications of quantum computers. Such simulations -- for which classical methods are often inaccurate -- are critical to advancing our knowledge and understanding of quantum chemistry and materials, underpinning a wide range of fields, from biochemistry to clean-energy technologies and chemical synthesis. However, the performance of all previous digital quantum simulations of fermions has been matched by classical methods, and it has thus far remained unclear whether near-term, intermediate-scale quantum hardware could offer any computational advantage in this area. Here, we implement an efficient quantum simulation algorithm on Quantinuum's System Model H2 trapped-ion quantum computer for the time dynamics of a 56-qubit system that is too complex for exact classical simulation. We focus on the periodic spinful 2D Fermi-Hubbard model and present evidence of spin-charge separation, where the elementary electron's charge and spin decouple. In the limited cases where ground truth is available through exact classical simulation, we find that it agrees with the results we obtain from the quantum device. Employing long-range Wilson operators to study deconfinement of the effective gauge field between spinons and the effective potential between charge carriers, we find behaviour that differs from predictions made by classical tensor network methods. Our results herald the use of quantum computing for simulating strongly correlated electronic systems beyond the capacity of classical computing.

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arXiv2025

Programmable digital quantum simulation of 2D Fermi-Hubbard dynamics using 72 superconducting qubits

Simulating the time-dynamics of quantum many-body systems was the original use of quantum computers proposed by Feynman, motivated by the critical role of quantum interactions between electrons in the properties of materials and molecules. Accurately simulating such systems remains one of the most promising applications of general-purpose digital quantum computers, in which all the parameters of the model can be programmed and any desired physical quantity output. However, performing such simulations on today's quantum computers at a scale beyond the reach of classical methods requires advances in the efficiency of simulation algorithms and error mitigation techniques. Here we demonstrate programmable digital quantum simulation of the dynamics of the 2D Fermi-Hubbard model -- one of the best-known simplified models of electrons in crystalline solids -- at a scale beyond exact classical state-vector simulation. We implement simulations of this model on lattice sizes up to ${6\times 6}$ using 72 qubits on Google's Willow quantum processor, across a range of physical parameters, including different on-site electron-electron interaction strengths and magnetic flux values, and study phenomena including formation of magnetic polarons (charge carriers surrounded by local magnetic polarisation), dynamical symmetry-breaking in stripe-ordered states, attraction of charge carriers on an entangled background state known as a valence bond solid, and the approach to equilibrium through thermalisation. We validate our results against exact calculations in parameter regimes where these are feasible, and compare them to approximate classical simulations performed using tensor network and operator propagation methods. Our results demonstrate that meaningful programmable digital quantum simulation of many-body interacting electron models is now feasible on state-of-the-art quantum hardware.

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arXiv2025

Quantum thermodynamics and semi-definite optimization

In quantum thermodynamics, a system is described by a Hamiltonian and a list of non-commuting charges representing conserved quantities like particle number or electric charge, and an important goal is to determine the system's minimum energy in the presence of these conserved charges. In optimization theory, a semi-definite program (SDP) involves a linear objective function optimized over the cone of positive semi-definite operators intersected with an affine space. These problems arise from differing motivations in the physics and optimization communities and are phrased using very different terminology, yet they are essentially identical mathematically. By adopting Jaynes' mindset motivated by quantum thermodynamics, we observe that minimizing free energy in the aforementioned thermodynamics problem, instead of energy, leads to an elegant solution in terms of a dual chemical potential maximization problem that is concave in the chemical potential parameters. As such, one can employ standard (stochastic) gradient ascent methods to find the optimal values of these parameters, and these methods are guaranteed to converge quickly. At low temperature, the minimum free energy provides an excellent approximation for the minimum energy. We then show how this Jaynes-inspired gradient-ascent approach can be used in both first- and second-order classical and hybrid quantum-classical algorithms for minimizing energy, and equivalently, how it can be used for solving SDPs, with guarantees on the runtimes of the algorithms. The approach discussed here is well grounded in quantum thermodynamics and, as such, provides physical motivation underpinning why algorithms published fifty years after Jaynes' seminal work, including the matrix multiplicative weights update method, the matrix exponentiated gradient update method, and their quantum algorithmic generalizations, perform well at solving SDPs.

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arXiv2025

Quantum natural gradient with thermal-state initialization

Parameterized quantum circuits (PQCs) are central to variational quantum algorithms (VQAs), yet their performance is hindered by complex loss landscapes that make their trainability challenging. Quantum natural gradient descent, which leverages the geometry of the parameterized space through quantum generalizations of the Fisher information matrix, offers a promising solution but has been largely limited to pure-state scenarios, with only approximate methods available for mixed-state settings. This paper addresses this question, originally posed in [Stokes et al., Quantum 4, 269 (2020)], by providing exact methods to compute three quantum generalizations of the Fisher information matrix-the Fisher-Bures, Wigner-Yanase, and Kubo-Mori information matrices-for PQCs initialized with thermal states. We prove that these matrix elements can be estimated using quantum algorithms combining the Hadamard test, classical random sampling, and Hamiltonian simulation. By broadening the set of quantum generalizations of Fisher information and realizing their unbiased estimation, our results enable the implementation of quantum natural gradient descent algorithms for mixed-state PQCs, thereby enhancing the flexibility of optimization when using VQAs. Another immediate consequence of our findings is to establish fundamental limitations on the ability to estimate the parameters of a state generated by an unknown PQC, when given sample access to such a state.

quant-ph↗

arXiv2025

Digital Quantum Simulations of the Non-Resonant Open Tavis-Cummings Model

The open Tavis--Cummings model consists of $N$ quantum emitters interacting with a common cavity mode, accounts for losses and decoherence, and is frequently explored for quantum information processing and designing quantum devices. As $N$ increases, it becomes harder to simulate the open Tavis--Cummings model using traditional methods. To address this problem, we implement two quantum algorithms for simulating the dynamics of this model in the inhomogeneous, non-resonant regime, with up to three excitations in the cavity. We show that the implemented algorithms have gate complexities that scale polynomially, as $O(N^2)$ and $O(N^3)$, while the number of qubits used by these algorithms (space complexity) scales linearly as $O(N)$. One of these algorithms is the sampling-based wave matrix Lindbladization algorithm, for which we propose two protocols to implement its system-independent fixed interaction, resolving key open questions of [Patel and Wilde, Open Sys. & Info. Dyn., 30:2350014 (2023)]. We benchmark our results against a classical differential equation solver in a variety of scenarios and demonstrate that our algorithms accurately reproduce the expected dynamics.

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arXiv2025

Evolved Quantum Boltzmann Machines

We introduce evolved quantum Boltzmann machines as a variational ansatz for quantum optimization and learning tasks. Given two parameterized Hamiltonians $G(\theta)$ and $H(\phi)$, an evolved quantum Boltzmann machine consists of preparing a thermal state of the first Hamiltonian $G(\theta)$ followed by unitary evolution according to the second Hamiltonian $H(\phi)$. Alternatively, one can think of it as first realizing imaginary time evolution according to $G(\theta)$ followed by real time evolution according to $H(\phi)$. After defining this ansatz, we provide analytical expressions for the gradient vector and illustrate their application in ground-state energy estimation and generative modeling, showing how the gradient for these tasks can be estimated by means of quantum algorithms that involve classical sampling, Hamiltonian simulation, and the Hadamard test. We also establish analytical expressions for the Fisher-Bures, Wigner-Yanase, and Kubo-Mori information matrix elements of evolved quantum Boltzmann machines, as well as quantum algorithms for estimating each of them, which leads to at least three different general natural gradient descent algorithms based on this ansatz. Along the way, we establish a broad generalization of the main result of [Luo, Proc. Am. Math. Soc. 132, 885 (2004)], proving that the Fisher-Bures and Wigner-Yanase information matrices of general parameterized families of states differ by no more than a factor of two in the matrix (Loewner) order, making them essentially interchangeable for training when using natural gradient descent.

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arXiv2024

Sample-based Hamiltonian and Lindbladian simulation: Non-asymptotic analysis of sample complexity

Density matrix exponentiation (DME) is a quantum algorithm that processes multiple copies of a program state $\sigma$ to realize the Hamiltonian evolution $e^{-i \sigma t}$. Wave matrix Lindbladization (WML) similarly processes multiple copies of a program state $\psi_L$ in order to realize a Lindbladian evolution. Both algorithms are prototypical sample-based quantum algorithms and can be used for various quantum information processing tasks, including quantum principal component analysis, Hamiltonian simulation, and Lindbladian simulation. In this work, we present detailed sample complexity analyses for DME and sample-based Hamiltonian simulation, as well as for WML and sample-based Lindbladian simulation. In particular, we prove that the sample complexity of DME is no larger than $4t^2/\varepsilon$ for evolution time $t$ and imprecision level $\varepsilon$ quantified by the normalized diamond distance. We also establish a fundamental lower bound on the sample complexity of sample-based Hamiltonian simulation, which matches our DME sample complexity bound up to a constant multiplicative factor. Additionally, we prove that the sample complexity of WML is no larger than $3t^2d^2/\varepsilon$, where $d$ is the dimension of the space on which the Lindblad operator acts nontrivially, and we prove a lower bound of $10^{-4} t^2/\varepsilon$ on the sample complexity of sample-based Lindbladian simulation. These results prove that WML is optimal for sample-based Lindbladian simulation whenever the Lindblad operator acts nontrivially on a constant-sized system. Finally, we point out that the DME sample complexity analysis in [Kimmel et al., npj Quantum Information 3, 13 (2017)] and the WML sample complexity analysis in [Patel and Wilde, Open Systems \& Information Dynamics 30, 2350010 (2023)] appear to be incomplete, highlighting the need for the results presented here.

quant-ph↗

arXiv2024

Natural gradient and parameter estimation for quantum Boltzmann machines

Thermal states play a fundamental role in various areas of physics, and they are becoming increasingly important in quantum information science, with applications related to semi-definite programming, quantum Boltzmann machine learning, Hamiltonian learning, and the related task of estimating the parameters of a Hamiltonian. Here we establish formulas underlying the basic geometry of parameterized thermal states, and we delineate quantum algorithms for estimating the values of these formulas. More specifically, we establish formulas for the Fisher--Bures and Kubo--Mori information matrices of parameterized thermal states, and our quantum algorithms for estimating their matrix elements involve a combination of classical sampling, Hamiltonian simulation, and the Hadamard test. These results have applications in developing a natural gradient descent algorithm for quantum Boltzmann machine learning, which takes into account the geometry of thermal states, and in establishing fundamental limitations on the ability to estimate the parameters of a Hamiltonian, when given access to thermal-state samples. For the latter task, and for the special case of estimating a single parameter, we sketch an algorithm that realizes a measurement that is asymptotically optimal for the estimation task. We finally stress that the natural gradient descent algorithm developed here can be used for any machine learning problem that employs the quantum Boltzmann machine ansatz.

quant-ph↗

arXiv2024

Quantum Boltzmann machine learning of ground-state energies

Estimating the ground-state energy of Hamiltonians is a fundamental task for which it is believed that quantum computers can be helpful. Several approaches have been proposed toward this goal, including algorithms based on quantum phase estimation and hybrid quantum-classical optimizers involving parameterized quantum circuits, the latter falling under the umbrella of the variational quantum eigensolver. Here, we analyze the performance of quantum Boltzmann machines for this task, which is a less explored ansatz based on parameterized thermal states and which is not known to suffer from the barren-plateau problem. We delineate a hybrid quantum-classical algorithm for this task and rigorously prove that it converges to an $\varepsilon$-approximate stationary point of the energy function optimized over parameter space, while using a number of parameterized-thermal-state samples that is polynomial in $\varepsilon^{-1}$, the number of parameters, and the norm of the Hamiltonian being optimized. Our algorithm estimates the gradient of the energy function efficiently by means of a quantum circuit construction that combines classical random sampling, Hamiltonian simulation, and the Hadamard test. Additionally, supporting our main claims are calculations of the gradient and Hessian of the energy function, as well as an upper bound on the matrix elements of the latter that is used in the convergence analysis.

quant-ph↗

arXiv2024

Optimal Coherent Quantum Phase Estimation via Tapering

Due to its significance as a subroutine, in this work, we consider the coherent version of the quantum phase estimation problem, where given an arbitrary input state and black-box access to unitaries $U$ and controlled-$U$, the goal is to estimate the phases of $U$ in superposition. Most existing phase estimation algorithms involve intermediary measurements that disrupt coherence. Only a couple of algorithms, including the standard quantum phase estimation algorithm, consider this coherent setting. However, the standard algorithm only succeeds with a constant probability. To boost this success probability, one can employ the coherent median technique, resulting in an algorithm with asymptotically optimal query complexity (the total number of calls to $U$ and controlled-$U$). However, this coherent median technique requires a large number of ancilla qubits and a computationally expensive quantum sorting network. To address this, in this work, we propose an improved version of the standard algorithm called the tapered quantum phase estimation (tQPE) algorithm, which leverages tapering (or window) functions commonly used in classical signal processing. Our algorithm achieves the asymptotically optimal query complexity without requiring the expensive coherent median technique to boost success probability. Moreover, we find the absolutely optimal taper - not only in the asymptotic scaling but in terms of exact performance. We provide an efficiently preparable ancilla state based on an approximation of the optimal taper, which incurs at most a factor-of-two increase in the probability of error, thereby maintaining near-optimal performance in practice. In the appendices, we give an explicit construction of the taper state preparation circuit. Finally, we derive an error bound for coherent QPE when the phase estimate is used as a control and subsequently uncomputed.

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arXiv2023

QSlack: A slack-variable approach for variational quantum semi-definite programming

Solving optimization problems is a key task for which quantum computers could possibly provide a speedup over the best known classical algorithms. Particular classes of optimization problems including semi-definite programming (SDP) and linear programming (LP) have wide applicability in many domains of computer science, engineering, mathematics, and physics. Here we focus on semi-definite and linear programs for which the dimensions of the variables involved are exponentially large, so that standard classical SDP and LP solvers are not helpful for such large-scale problems. We propose the QSlack and CSlack methods for estimating their optimal values, respectively, which work by 1) introducing slack variables to transform inequality constraints to equality constraints, 2) transforming a constrained optimization to an unconstrained one via the penalty method, and 3) replacing the optimizations over all possible non-negative variables by optimizations over parameterized quantum states and parameterized probability distributions. Under the assumption that the SDP and LP inputs are efficiently measurable observables, it follows that all terms in the resulting objective functions are efficiently estimable by either a quantum computer in the SDP case or a quantum or probabilistic computer in the LP case. Furthermore, by making use of SDP and LP duality theory, we prove that these methods provide a theoretical guarantee that, if one could find global optima of the objective functions, then the resulting values sandwich the true optimal values from both above and below. Finally, we showcase the QSlack and CSlack methods on a variety of example optimization problems and discuss details of our implementation, as well as the resulting performance. We find that our implementations of both the primal and dual for these problems approach the ground truth, typically achieving errors of order $10^{-2}$.

quant-ph↗

arXiv2023

Dual-VQE: A quantum algorithm to lower bound the ground-state energy

The variational quantum eigensolver (VQE) is a hybrid quantum-classical variational algorithm that produces an upper-bound estimate of the ground-state energy of a Hamiltonian. As quantum computers become more powerful and go beyond the reach of classical brute-force simulation, it is important to assess the quality of solutions produced by them. Here we propose a dual variational quantum eigensolver (dual-VQE) that produces a lower-bound estimate of the ground-state energy. As such, VQE and dual-VQE can serve as quality checks on their solutions; in the ideal case, the VQE upper bound and the dual-VQE lower bound form an interval containing the true optimal value of the ground-state energy. The idea behind dual-VQE is to employ semidefinite programming duality to rewrite the ground-state optimization problem as a constrained maximization problem, which itself can be bounded from below by an unconstrained optimization problem to be solved by a variational quantum algorithm. When using a convex combination ansatz in conjunction with a classical generative model, the quantum computational resources needed to evaluate the objective function of dual-VQE are no greater than those needed for that of VQE. We also show that the problem is well suited for classical pretraining using matrix product states and these methods help warm-start the optimization. We simulated the performance of dual-VQE on the transverse-field Ising model with and without pretraining and found that, for the example considered, while dual-VQE training is slower and noisier than VQE, it approaches the true value with an error of order $10^{-2}$.

quant-ph↗

arXiv2023

Wave Matrix Lindbladization II: General Lindbladians, Linear Combinations, and Polynomials

In this paper, we investigate the problem of simulating open system dynamics governed by the well-known Lindblad master equation. In our prequel paper, we introduced an input model in which Lindblad operators are encoded into pure quantum states, called program states, and we also introduced a method, called wave matrix Lindbladization, for simulating Lindbladian evolution by means of interacting the system of interest with these program states. Therein, we focused on a simple case in which the Lindbladian consists of only one Lindblad operator and a Hamiltonian. Here, we extend the method to simulating general Lindbladians and other cases in which a Lindblad operator is expressed as a linear combination or a polynomial of the operators encoded into the program states. We propose quantum algorithms for all these cases and also investigate their sample complexity, i.e., the number of program states needed to simulate a given Lindbladian evolution approximately. Finally, we demonstrate that our quantum algorithms provide an efficient route for simulating Lindbladian evolution relative to full tomography of encoded operators, by proving that the sample complexity for tomography is dependent on the dimension of the system, whereas the sample complexity of wave matrix Lindbladization is dimension independent.

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arXiv2023

Wave Matrix Lindbladization I: Quantum Programs for Simulating Markovian Dynamics

Density Matrix Exponentiation is a technique for simulating Hamiltonian dynamics when the Hamiltonian to be simulated is available as a quantum state. In this paper, we present a natural analogue to this technique, for simulating Markovian dynamics governed by the well known Lindblad master equation. For this purpose, we first propose an input model in which a Lindblad operator $L$ is encoded into a quantum state $\psi$. Then, given access to $n$ copies of the state $\psi$, the task is to simulate the corresponding Markovian dynamics for time $t$. We propose a quantum algorithm for this task, called Wave Matrix Lindbladization, and we also investigate its sample complexity. We show that our algorithm uses $n = O(t^2/\varepsilon)$ samples of $\psi$ to achieve the target dynamics, with an approximation error of $O(\varepsilon)$.

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