arXiv subjects

Aram W. Harrow

Publications and source records attributed to Aram W. Harrow.

At least 19 recordsLinked to original sources

arXiv2026

Randomized truncation of quantum states

A fundamental task in quantum information is to approximate a pure quantum state in terms of sparse states or, for a bipartite system, states of bounded Schmidt rank. The optimal deterministic approximation in each case is straightforward, and maximizes the fidelity: keep the largest entries or singular values. On the other hand, random mixtures of sparse states can achieve quadratically improved trace distances, and yield nontrivial bounds on other distance measures like the robustness. In this work, we give efficient algorithms for finding mixtures of sparse states that optimally approximate a given pure state in either trace distance or robustness. These algorithms also yield descriptions of efficiently samplable ensembles of sparse, or less-entangled, states that correspond to these optimal mixed approximations. This can be used for the truncation step of algorithms for matrix product states, improving their accuracy while using no extra memory and no asymptotic increase in runtime, and we demonstrate this improvement numerically. Our proofs rely on tools from convex optimization and zero-sum games, as well as rigorous gurarantees for computing and sampling from maximum-entropy distributions. In particular, a key step in our construction is conditional Poisson sampling, which is the well-studied problem of drawing fixed-size subsets from a population with known inclusion probabilities. We give improved algorithms for carrying out this sampling, which may be of independent interest.

quant-ph↗

arXiv2026

Sublinear-depth Quantum Simulation of Electrons with Atomic Orbitals

Discretizing electronic degrees of freedom into atomic orbitals is the default choice to accurately model electrons in molecules in a compact and flexible way. We consider the task of simulating such electronic structure models on a quantum computer for $N$ orbitals, keeping the number of orbitals per atom constant. Atomic orbitals give rise to Hamiltonians with potentially up to $\mathcal{O}(N^4)$ terms and little structure, which made it challenging to match the complexity of methods based on plane waves and real-space grids. In this work, we show that a Trotter step for atomic orbitals can be implemented in depth $\mathcal{O}(\mathrm{polylog}(N))$ without ancillas by combining a rigorous treatment of orbital localization, a hierarchical Hamiltonian decomposition based on the Fast Multipole Method, and shallow quantum Fourier arithmetic circuits. By bounding the Trotter error, we obtain sublinear simulation depth, and our total gate count $N^{5/3 + o(1)}$ matches the best scaling known for any basis.

quant-ph↗

arXiv2026

An Exponential Sample-Complexity Advantage for Coherent Quantum Inference

Standard quantum inference converts quantum data into classical outputs. We study an alternative inference setting in which the desired output is quantum. Such settings include quantum purity amplification (QPA), random purification and approximate cloning, and density matrix exponentiation. We show that such protocols can achieve exponentially lower sample complexity than incoherent, measurement-mediated protocols, establishing, to the best of our knowledge, the first nontrivial separations between dimension-independent coherent and dimension-dependent incoherent sample complexity. For QPA with principal-eigenstate targets and $d$-dimensional inputs, at fixed output number and spectral gap, coherent processing achieves error $\varepsilon$ using $O(1/\varepsilon)$ copies, whereas any incoherent protocol requires $Ω(d/\varepsilon)$ copies. Together, these separations motivate a unified picture of coherent quantum inference, in which an entanglement-breaking limit identifies the optimal incoherent counterpart.

quant-ph↗

arXiv2026

Quantum Purity Amplification for Arbitrary Eigenstates and Multiple Outputs

Quantum purity amplification (QPA) is the task of coherently transforming $n$ copies of a mixed state into high-fidelity copies of a chosen eigenstate. We develop a general theory of QPA for $n$ input copies, $m$ output copies, and arbitrary choices of target eigenstate, local dimension $d$, and input spectrum. We introduce the overhang-removal protocol, derive its all-site and one-site fidelities across output regimes, and show that it is optimal in several regimes, both asymptotically and nonasymptotically. For the asymptotic analysis, we use a path-graph parametrization to show that, when the target eigenvalue has a constant spectral gap $D_{k,\mathrm{min}}$, achieving worst-case all-site error $\varepsilon$ scales as $O(m/(\varepsilon D_{k,\mathrm{min}}^2))$. When $m/n$ approaches a constant, the performance exhibits phase-like regimes, which we characterize explicitly. For the nonasymptotic analysis, we develop a theory of generalized Young diagrams that yields tight sample complexity bounds and provides the first tight dimension-uniform guarantee for optimal QPA. We also provide asymptotically efficient implementations of the overhang-removal protocol. Together, these results establish QPA as a rigorous example of coherent quantum information processing with dimension-uniform sample complexity, supplying the technical foundation for the coherent-incoherent separation developed in the companion work.

quant-ph↗

arXiv2026

An Optimal Quantum Linear Systems Algorithm

In the quantum linear systems problem (QLSP), we are given query access to a $d$-sparse $N\times N$ matrix $A$ with condition number $κ$, and the ability to prepare a quantum state proportional to a vector $\vec b$. The goal is to output an $ε$-approximation to the quantum state proportional to the solution $\vec x$ of $A\vec{x}=\vec{b}$. Following a long line of work, the best previously known quantum algorithms for the QLSP had query complexities $O(κd\log(1/ε))$ and $κ\sqrt d(κd/ε)^{o(1)}$, while the best known lower bounds were $Ω(κ\log(1/ε))$ and $Ω(κ\sqrt d)$. We improve these bounds and show that the complexity of the QLSP is $Θ(κ\sqrt d\log(1/ε))$. We also resolve an open problem of Berry and Childs by showing that any $N\times N$ unitary can be implemented with bounded error using $O(\sqrt N)$ queries to its matrix entries.

quant-ph↗

arXiv2026

Fast Hamiltonian engineering from cut polytope geometry

Hamiltonian engineering (HE) simulates quantum dynamics under a target Hamiltonian using a native entangling system Hamiltonian and restricted control, with applications from quantum gate design to analog quantum simulation. Observing that, for many systems, formulating HE as a linear program only requires specific commutation relations between the control pulses and the system Hamiltonian, we provide a unified framework for broad classes of $2$-local qubit, qudit, and fermionic systems. We reformulate time-optimal HE as a complex $k$-cut polytope problem, with $k$ the number of distinct phases in these commutation relations, and prove it NP-complete for every finite $k$. We therefore relax the polytope to the elliptope and apply a Krivine-type rounding, yielding pulses informed by the system and target Hamiltonians, which we then use in a linear program. We derive how many informed pulses are necessary and sufficient for the linear program to be reliably feasible. Additionally mixing in uniformly sampled pulses guarantees a solution close to the relaxation value from $\mathrm{O}(m)$ pulses, with $m$ the number of interaction terms. Together with upper and lower bounds on the optimal quantum run time, this yields an $\mathrm{O}(\sqrt{m})$ approximation ratio. In benchmarks on a fully connected Ising model, a chiral clock model for qudits, and the fermionic Harper-Hofstadter model, our approach attains near-optimal quantum run times wherever the optimum is computable, reaches run times that saturate with the lattice size in the fermionic case, and outperforms state-of-the-art methods with $\mathrm{O}(m)$ pulses. This establishes a unified approach with provable guarantees to the automatic programming of analog quantum simulators.

quant-ph↗

arXiv2026

Depth-1 expanders on the unitary group and applications

We construct a constant-degree and constant-gap quantum expander on $n$ qubits where each unitary can be implemented by a depth-$1$ and 1D circuit of Pauli or CNOT gates. We provide two applications of this expander. First, we use it to construct a family of frustration-free 1D Hamiltonians whose ground states obey the entanglement-gap relation $S = Θ(Δ^{-1/2})$; this is believed to be optimal, but achieving it had been open. Second, we use it to provide a streaming protocol that tests for closeness to a class of 1D volume-law entangled states. Moreover, we extend our quantum expander to a constant-degree and constant-gap expander on the unitary group where each unitary is a single $T$ gate, a single $T^{\dagger}$ gate, or a depth-$1$ Clifford circuit. This implies that a random sequence of unitaries from the expander yields a gapped walk on a dense subgroup of the unitary group. This improves upon previous work by Bourgain and Gamburd which did not control the dependence of the gap on the dimension.

quant-ph↗

arXiv2026

Strategic Plan for Neutral Atom Quantum Computation

We present a strategic plan for neutral atom quantum computation, bringing together hardware development and theory advancements to achieve the goal of practical quantum advantage. The concept of practical quantum advantage is defined, along with how to verify claims of advantage, and approaches to designing quantum algorithms that deliver practical advantage. Future directions for neutral atom quantum processor hardware are described: scaling-up system size, Qubit encodings and atomic platforms, going further below threshold with neutral-atom logical-qubit performance, continuous reloading of qubits, and fast readout. We also explore opportunities for scalable integrated photonic control technologies. Alongside hardware advancements, new developments in quantum error correction and compilation of quantum circuits are proposed. Finally, we examine the opportunity of networking multiple neutral atom quantum processors together to perform distributed quantum computing and overcome possible limitations of a single system.

quant-ph↗

arXiv2026

Optimal Lower Bounds for Hamiltonian Simulation

For Hamiltonian $H = \sum_j h_j$, we prove asymptotically tight lower bounds on the gate and query complexities of simulating time evolution on a quantum computer. Our bounds hold for arbitrary term norms $\|h_j\|$, time $t$, and trace-distance error $ε$. The matching upper bound (known as composite qDRIFT) consists of high-order Trotterization of the large terms and a randomized first-order Trotterization of the small terms. Unlike prior work that chooses worst-case $\|h_j\|$ to encode the computation of parity or other Boolean functions in time evolution, our proof is elementary and based on a local, bounded-degree classical Hamiltonian. Our work suggests that for many physical systems (e.g., power-law interactions), gate count must scale polynomially in $1/ε$, contrary to the complexity suggested by counting coherent oracle queries such as those in the block-encoding model.

quant-ph↗

arXiv2026

Plethysm is in #BQP

Some representation-theoretic multiplicities, such as the Kostka and the Littlewood-Richardson coefficients, admit a combinatorial interpretation that places their computation in the complexity class #P. Whether this holds more generally is considered an important open problem in mathematics and computer science, with relevance for geometric complexity theory and quantum information. Recent work has investigated the quantum complexity of particular multiplicities, such as the Kronecker coefficients and certain special cases of the plethysm coefficients. Here, we show that a broad class of representation-theoretic multiplicities is in #BQP. In particular, our result implies that the plethysm coefficients are in #BQP, which was only known in special cases. It also implies all known results on the quantum complexity of previously studied coefficients as special cases, unifying, simplifying, and extending prior work. We obtain our result by multiple applications of the Schur transform. Recent work has improved its dependence on the local dimension, which is crucial for our work. We further describe a general approach for showing that representation-theoretic multiplicities are in #BQP that captures our approach as well as the approaches of prior work. We complement the above by showing that the same multiplicities are also naturally in GapP and obtain polynomial-time classical algorithms when certain parameters are fixed.

quant-ph↗

arXiv2025

TEPID-ADAPT: Adaptive variational method for simultaneous preparation of low-temperature Gibbs and low-lying eigenstates

Preparing Gibbs states, which describe systems in equilibrium at finite temperature, is of great importance, particularly at low temperatures. In this work, we propose a new method -- TEPID-ADAPT -- that prepares the thermal Gibbs state of a given Hamiltonian at low temperatures using a variational method that is partially adaptive and uses a purification with a minimal number of ancillary qubits. We also present an alternative implementation without ancillary qubits. A key technical innovation here is to use a mixed-state ansatz where the entropy can be efficiently calculated, with no computational overhead. Our algorithm uses a truncated, parametrized eigenspectrum of the Hamiltonian. Beyond preparing Gibbs states, this approach also straightforwardly gives us access to the truncated low-energy eigenspectrum of the Hamiltonian, making it also a method that prepares excited states simultaneously. As a result of this, we are also able to prepare thermal states at any lower temperature of the same Hamiltonian without further optimization.

quant-ph↗

arXiv2025

Toward Quantum-Enabled Biomarker Discovery: An Outlook from Q4Bio

We present a case study and forward-looking perspective on co-design for hybrid quantum-classical algorithms, centered on the goal of empirical quantum advantage (EQA), which we define as a measurable performance gain using quantum hardware over state-of-the-art classical methods on the same task. Because classical algorithms continue to improve, the EQA crossover point is a moving target; nevertheless, we argue that a persistent advantage is possible for our application class even if the crossover point shifts. Specifically, our team examines the task of biomarker discovery in precision oncology. We push the limitations of the best classical algorithms, improving them as best as we can, and then augment them with a quantum subroutine for the task where we are most likely to see performance gains. We discuss the implementation of a quantum subroutine for feature selection on current devices, where hardware constraints necessitate further co-design between algorithm and physical device capabilities. Looking ahead, we perform resource analysis to explore a plausible EQA region on near/intermediate-term hardware, considering the impacts of advances in classical and quantum computing on this regime. Finally, we outline potential clinical impact and broader applications of this hybrid pipeline beyond oncology.

quant-ph↗

arXiv2025

How to use quantum computers for biomolecular free energies

Free energy calculations are at the heart of physics-based analyses of biochemical processes. They allow us to quantify molecular recognition mechanisms, which determine a wide range of biological phenomena from how cells send and receive signals to how pharmaceutical compounds can be used to treat diseases. Quantitative and predictive free energy calculations require computational models that accurately capture both the varied and intricate electronic interactions between molecules as well as the entropic contributions from motions of these molecules and their aqueous environment. However, accurate quantum-mechanical energies and forces can only be obtained for small atomistic models, not for large biomacromolecules. Here, we demonstrate how to consistently link accurate quantum-mechanical data obtained for substructures to the overall potential energy of biomolecular complexes by machine learning in an integrated algorithm. We do so using a two-fold quantum embedding strategy where the innermost quantum cores are treated at a very high level of accuracy. We demonstrate the viability of this approach for the molecular recognition of a ruthenium-based anticancer drug by its protein target, applying traditional quantum chemical methods. As such methods scale unfavorable with system size, we analyze requirements for quantum computers to provide highly accurate energies that impact the resulting free energies. Once the requirements are met, our computational pipeline FreeQuantum is able to make efficient use of the quantum computed energies, thereby enabling quantum computing enhanced modeling of biochemical processes. This approach combines the exponential speedups of quantum computers for simulating interacting electrons with modern classical simulation techniques that incorporate machine learning to model large molecules.

quant-ph↗

arXiv2025

Future of Quantum Computing

On Tuesday 26th November 2024, four discussants participated in a moderated virtual panel titled Future of Quantum Computing as one session of the 8th International Conference on Quantum Techniques in Machine Learning hosted by the University of Melbourne. This article provides a detailed summary of the discussion in this lively session.

quant-ph↗

arXiv2025

Quantum entropy thermalization

In an isolated quantum many-body system undergoing unitary evolution, the entropy of a subsystem (smaller than half the system size) thermalizes if at long times, it is to leading order equal to the thermodynamic entropy of the subsystem at the same energy. In this paper, we prove entropy thermalization for a nearly integrable Sachdev-Ye-Kitaev model initialized in a pure product state. The model is obtained by adding random all-to-all $4$-body interactions as a perturbation to a random free-fermion model.

cond-mat.stat-mech↗

arXiv2024

Optimal quantum circuit cuts with application to clustered Hamiltonian simulation

We study methods to replace entangling operations with random local operations in a quantum computation, at the cost of increasing the number of required executions. First, we consider "space-like cuts" where an entangling unitary is replaced with random local unitaries. We propose an entanglement measure for quantum dynamics, the product extent, which bounds the cost in a procedure for this replacement based on two copies of the Hadamard test. In the terminology of prior work, this procedure yields a quasiprobability decomposition with minimal 1-norm in a number of cases, which addresses an open question of Piveteau and Sutter. As an application, we give an improved algorithm for clustered Hamiltonian simulation. Specifically we show that interactions can be removed at a cost which is exponential in the sum of their strengths times the evolution time, and vanishing in the limit of weak interactions. We also give an improved upper bound on the cost of replacing wires with measure-and-prepare channels using "time-like cuts''. We prove a matching information-theoretic lower bound when estimating output probabilities.

quant-ph↗

arXiv2023

Thermalization without eigenstate thermalization

In an isolated quantum many-body system undergoing unitary evolution, we study the thermalization of a subsystem, treating the rest of the system as a bath. In this setting, the eigenstate thermalization hypothesis (ETH) was proposed to explain thermalization. Consider a nearly integrable Sachdev-Ye-Kitaev model obtained by adding random all-to-all 4-body interactions as a perturbation to a random free-fermion model. When the subsystem size is larger than the square root of but is still a vanishing fraction of the system size, we prove thermalization if the system is initialized in a random product state, while almost all eigenstates violate the ETH. In this sense, the ETH is not a necessary condition for thermalization.

cond-mat.stat-mech↗

arXiv2023

Improved concentration of Laguerre and Jacobi ensembles

We consider the asymptotic limits where certain parameters in the definitions of the Laguerre and Jacobi ensembles diverge. In these limits, Dette, Imhof, and Nagel proved that up to a linear transformation, the joint probability distributions of the ensembles become more and more concentrated around the zeros of the Laguerre and Jacobi polynomials, respectively. In this paper, we improve the concentration bounds. Our proofs are similar to those in the original references, but the error analysis is improved and arguably simpler. For the first and second moments of the Jacobi ensemble, we further improve the concentration bounds implied by our aforementioned results.

math.PR↗