arXiv subjects

Aram W. Harrow

Publications and source records attributed to Aram W. Harrow.

At least 19 recordsLinked to original sources

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 $S = \Theta(\Delta^{-1/2})$; 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 $\epsilon$. 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 $1/\epsilon$, contrary to the complexity suggested by counting coherent oracle queries such as those in the block-encoding model.

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, preserving coherence. Such settings include quantum purity amplification (QPA), mixed-state approximate purification or cloning, and density matrix exponentiation. We show that such protocols can achieve exponentially lower sample complexity than incoherent, measurement-mediated protocols. For QPA with principal eigenstate targets and $d$-dimensional inputs, coherent processing achieves error $\varepsilon$ using $O(1/\varepsilon)$ copies, versus the $\Omega(d/\varepsilon)$ copies required by any incoherent protocol. Together, these sharp coherent-incoherent separations seed a theory of coherent quantum inference, with an entanglement-breaking limit identifying the optimal incoherent counterpart of each coherent protocol.

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 solve QPA in the general setting of $n$ input copies, $m$ output copies, arbitrary target eigenstates, arbitrary local dimension $d$, and generic input spectra. We characterize the optimal channel and derive its all-site and one-site performance laws across output regimes. 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 all-site error $\varepsilon$ requires a number of input copies independent of $d$ and scaling 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 dimension-uniform guarantee for optimal QPA. We also provide asymptotically efficient implementations of the optimal 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

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

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 we demonstrate this improvement numerically. Our proofs use basic facts about convex optimization and zero-sum games, as well as rigorous guarantees for computing maximum-entropy distributions.

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

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↗

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

Approximate orthogonality of permutation operators, with application to quantum information

Consider the $n!$ different unitary matrices that permute $n$ $d$-dimensional quantum systems. If $d\geq n$ then they are linearly independent. This paper discusses a sense in which they are approximately orthogonal (with respect to the Hilbert-Schmidt inner product) if $d\gg n^2$, or, in a different sense, if $d\gg n$. Previous work had shown pairwise approximate orthogonality of these matrices, but here we show a more collective statement, quantified in terms of the operator norm distance of the Gram matrix to the identity matrix. This simple point has several applications in quantum information and random matrix theory: (1) showing that random maximally entangled states resemble fully random states, (2) showing that Boson sampling output probabilities resemble those from Gaussian matrices, (3) improving the Eggeling-Werner scheme for multipartite data hiding, (4) proving that the product test of Harrow-Montanaro cannot be performed using LOCC without a large number of copies of the state to be tested, (5) proving that the purity of a quantum state also cannot be efficiently tested using LOCC, and (6, published separately) helping prove that poly-size random quantum circuits are poly-designs.

quant-ph↗

arXiv2023

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↗

arXiv2022

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↗

arXiv2022

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↗

arXiv2022

Maximal entanglement velocity implies dual unitarity

A global quantum quench can be modeled by a quantum circuit with local unitary gates. In general, entanglement grows linearly at a rate given by entanglement velocity, which is upper bounded by the growth of the light cone. We show that the unitary interactions achieving the maximal rate must remain unitary if we exchange the space and time directions -- a property known as dual unitarity. Our results are robust: approximate maximal entanglement velocity also implies approximate dual unitarity. We further show that maximal entanglement velocity is always accompanied by a specific dynamical pattern of entanglement, which yields simpler analyses of several known exactly solvable models.

quant-ph↗

arXiv2020

From communication complexity to an entanglement spread area law in the ground state of gapped local Hamiltonians

In this work, we make a connection between two seemingly different problems. The first problem involves characterizing the properties of entanglement in the ground state of gapped local Hamiltonians, which is a central topic in quantum many-body physics. The second problem is on the quantum communication complexity of testing bipartite states with EPR assistance, a well-known question in quantum information theory. We construct a communication protocol for testing (or measuring) the ground state and use its communication complexity to reveal a new structural property for the ground state entanglement. This property, known as the entanglement spread, roughly measures the ratio between the largest and the smallest Schmidt coefficients across a cut in the ground state. Our main result shows that gapped ground states possess limited entanglement spread across any cut, exhibiting an "area law" behavior. Our result quite generally applies to any interaction graph with an improved bound for the special case of lattices. This entanglement spread area law includes interaction graphs constructed in [Aharonov et al., FOCS'14] that violate a generalized area law for the entanglement entropy. Our construction also provides evidence for a conjecture in physics by Li and Haldane on the entanglement spectrum of lattice Hamiltonians [Li and Haldane, PRL'08]. On the technical side, we use recent advances in Hamiltonian simulation algorithms along with quantum phase estimation to give a new construction for an approximate ground space projector (AGSP) over arbitrary interaction graphs.

quant-ph↗