arXiv subjects

David Gosset

Publications and source records attributed to David Gosset.

At least 19 recordsLinked to original sources

arXiv2026

Conditional dependence and Scrooge ensembles in shallow random quantum circuits

The output state of a 2D geometrically local shallow random quantum circuit does not have long range correlations due to its lightcone structure. But this changes if one measures a subset of the qubits: long-range entanglement can be induced by the measurement process, leading to conditional correlations between distant qubits. In this paper we investigate the structure of conditional dependence in these circuits and its consequences for quantum advantage. For a tripartition $ABC$ of the qubits, we consider the ensemble of post-measurement states on $A$ that is conditioned on a specific measurement outcome on $B$ and ranges over all possible measurement outcomes on $C$. For circuit depths exceeding a constant critical value $d^*$, we conjecture that this ensemble is well approximated by a certain generalization of the Haar ensemble, called the Scrooge ensemble~[Jozsa \textit{et al.}, \href{https://doi.org/10.1103/PhysRevA.49.668}{Phys. Rev. A \textbf{49}, 668 (1994)}]; we also provide supporting numerical and analytical evidence. Our conjecture describes a precise sense in which the state retains its lightcone structure on the remaining unmeasured qubits, but also develops some globally random features arising from the measurement. A consequence is that $n$-qubit shallow random quantum circuits in two dimensions are classically efficiently simulable in the presence of a tiny depolarizing noise rate $\Omega(\log(n)/n)$.

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arXiv2026

Efficient quantum algorithm for Heisenberg spin systems

We consider a broad family of Heisenberg-type quantum spin systems that were studied by Suzuki and Fisher in 1971. This family includes, for example, the Heisenberg antiferromagnet on any bipartite graph. The ground and Gibbs states of these models are Lee-Yang tensors with radius 1: they are associated with multilinear polynomials that possess an extraordinary zero-freeness property inside the unit polydisk in the complex plane. For each Hamiltonian in this family we show that the spectral gap between the first-excited and ground-state energies is lower bounded by $2\mu$, where $\mu$ is the magnetic field along the $Z$ direction. Using this result we obtain an efficient quantum adiabatic algorithm for the ground energy of any model in this family. The proof is based on a new inequality that relates the spectral gap of a positive semidefinite operator to its Lee-Yang radius -- a quantitative strengthening of prior work of the authors that may find applications elsewhere.

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arXiv2026

Lee-Yang tensors and Hamiltonian complexity

A complex tensor with $n$ binary indices can be identified with a multilinear polynomial in $n$ complex variables. We say it is a Lee-Yang tensor with radius $r$ if the polynomial is nonzero whenever all variables lie in the open disk of radius $r$. In this work we study quantum states and observables which are Lee-Yang tensors when expressed in the computational basis. We first review their basic properties, including closure under tensor contraction and certain quantum operations. We show that quantum states with Lee-Yang radius $r > 1$ can be prepared by quasipolynomial-sized circuits. We also show that every Hermitian operator with Lee-Yang radius $r > 1$ has a unique principal eigenvector. These results suggest that $r = 1$ is a key threshold for quantum states and observables. Finally, we consider a family of two-local Hamiltonians where every interaction term energetically favors a deformed EPR state $|00\rangle + s|11\rangle$ for some $0 \leq s \leq 1$. We numerically investigate this model and find that on all graphs considered the Lee-Yang radius of the ground state is at least $r = 1/\sqrt{s}$ while the spectral gap between the two smallest eigenvalues is at least $1-s^2$. We conjecture that these lower bounds hold more generally; in particular, this would provide an efficient quantum adiabatic algorithm for the quantum Max-Cut problem on uniformly weighted bipartite graphs.

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arXiv2025

Multi-qubit Toffoli with exponentially fewer T gates

Prior work of Beverland et al. has shown that any exact Clifford+$T$ implementation of the $n$-qubit Toffoli gate must use at least $n$ $T$ gates. Here we show how to get away with exponentially fewer $T$ gates, at the cost of incurring a tiny $1/\mathrm{poly}(n)$ error that can be neglected in most practical situations. More precisely, the $n$-qubit Toffoli gate can be implemented to within error $\epsilon$ in the diamond distance by a randomly chosen Clifford+$T$ circuit with at most $O(\log(1/\epsilon))$ $T$ gates. We also give a matching $\Omega(\log(1/\epsilon))$ lower bound that establishes optimality, and we show that any purely unitary implementation achieving even constant error must use $\Omega(n)$ $T$ gates. We also extend our sampling technique to implement other Boolean functions. Finally, we describe upper and lower bounds on the $T$-count of Boolean functions in terms of non-adaptive parity decision tree complexity and its randomized analogue.

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arXiv2025

Classical and quantum algorithms for characters of the symmetric group

Characters of irreducible representations are ubiquitous in group theory. However, computing characters of some groups such as the symmetric group $S_n$ is a challenging problem known to be $\#P$-hard in the worst case. Here we describe a Matrix Product State (MPS) algorithm for characters of $S_n$. The algorithm computes an MPS encoding all irreducible characters of a given permutation. It relies on a mapping from characters of $S_n$ to quantum spin chains proposed by Crichigno and Prakash. We also provide a simpler derivation of this mapping. We complement this result by presenting a $poly(n)$ size quantum circuit that prepares the corresponding MPS, obtaining an efficient quantum algorithm for certain sampling problems based on characters of $S_n$. To assess classical hardness of these problems we present a general reduction from strong simulation (computing a given probability) to weak simulation (sampling with a small error). This reduction applies to any sampling problem with a certain granularity structure and may be of independent interest.

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arXiv2024

Quantum state preparation with optimal T-count

How many T gates are needed to approximate an arbitrary $n$-qubit quantum state to within error $\varepsilon$? Improving prior work of Low, Kliuchnikov, and Schaeffer, we show that the optimal asymptotic scaling is $\Theta\left(\sqrt{2^n\log(1/\varepsilon)}+\log(1/\varepsilon)\right)$ if we allow ancilla qubits. We also show that this is the optimal T-count for implementing an arbitrary diagonal $n$-qubit unitary to within error $\varepsilon$. We describe applications in which a tensor product of many single-qubit unitaries can be synthesized in parallel for the price of one.

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arXiv2024

Quantum advantage from measurement-induced entanglement in random shallow circuits

We study random constant-depth quantum circuits in a two-dimensional architecture. While these circuits only produce entanglement between nearby qubits on the lattice, long-range entanglement can be generated by measuring a subset of the qubits of the output state. It is conjectured that this long-range measurement-induced entanglement (MIE) proliferates when the circuit depth is at least a constant critical value. For circuits composed of Haar-random two-qubit gates, it is also believed that this coincides with a quantum advantage phase transition in the classical hardness of sampling from the output distribution. Here we provide evidence for a quantum advantage phase transition in the setting of random Clifford circuits. Our work extends the scope of recent separations between the computational power of constant-depth quantum and classical circuits, demonstrating that this kind of advantage is present in canonical random circuit sampling tasks. In particular, we show that in any architecture of random shallow Clifford circuits, the presence of long-range MIE gives rise to an unconditional quantum advantage. In contrast, any depth-d 2D quantum circuit that satisfies a short-range MIE property can be classically simulated efficiently and with depth O(d). Finally, we introduce a two-dimensional, depth-2, "coarse-grained" circuit architecture, composed of random Clifford gates acting on O(log n) qubits, for which we prove the existence of long-range MIE and establish an unconditional quantum advantage.

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arXiv2024

Triply efficient shadow tomography

Given copies of a quantum state $\rho$, a shadow tomography protocol aims to learn all expectation values from a fixed set of observables, to within a given precision $\epsilon$. We say that a shadow tomography protocol is triply efficient if it is sample- and time-efficient, and only employs measurements that entangle a constant number of copies of $\rho$ at a time. The classical shadows protocol based on random single-copy measurements is triply efficient for the set of local Pauli observables. This and other protocols based on random single-copy Clifford measurements can be understood as arising from fractional colorings of a graph $G$ that encodes the commutation structure of the set of observables. Here we describe a framework for two-copy shadow tomography that uses an initial round of Bell measurements to reduce to a fractional coloring problem in an induced subgraph of $G$ with bounded clique number. This coloring problem can be addressed using techniques from graph theory known as chi-boundedness. Using this framework we give the first triply efficient shadow tomography scheme for the set of local fermionic observables, which arise in a broad class of interacting fermionic systems in physics and chemistry. We also give a triply efficient scheme for the set of all $n$-qubit Pauli observables. Our protocols for these tasks use two-copy measurements, which is necessary: sample-efficient schemes are provably impossible using only single-copy measurements. Finally, we give a shadow tomography protocol that compresses an $n$-qubit quantum state into a $\text{poly}(n)$-sized classical representation, from which one can extract the expected value of any of the $4^n$ Pauli observables in $\text{poly}(n)$ time, up to a small constant error.

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arXiv2023

Classical simulation of peaked shallow quantum circuits

An $n$-qubit quantum circuit is said to be peaked if it has an output probability that is at least inverse-polynomially large as a function of $n$. We describe a classical algorithm with quasipolynomial runtime $n^{O(\log{n})}$ that approximately samples from the output distribution of a peaked constant-depth circuit. We give even faster algorithms for circuits composed of nearest-neighbor gates on a $D$-dimensional grid of qubits, with polynomial runtime $n^{O(1)}$ if $D=2$ and almost-polynomial runtime $n^{O(\log{\log{n}})}$ for $D>2$. Our sampling algorithms can be used to estimate output probabilities of shallow circuits to within a given inverse-polynomial additive error, improving previously known methods. As a simple application, we obtain a quasipolynomial algorithm to estimate the magnitude of the expected value of any Pauli observable in the output state of a shallow circuit (which may or may not be peaked). This is a dramatic improvement over the prior state-of-the-art algorithm which had an exponential scaling in $\sqrt{n}$.

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arXiv2023

Quantum complexity of the Kronecker coefficients

Whether or not the Kronecker coefficients of the symmetric group count some set of combinatorial objects is a longstanding open question. In this work we show that a given Kronecker coefficient is proportional to the rank of a projector that can be measured efficiently using a quantum computer. In other words a Kronecker coefficient counts the dimension of the vector space spanned by the accepting witnesses of a QMA verifier, where QMA is the quantum analogue of NP. This implies that approximating the Kronecker coefficients to within a given relative error is not harder than a certain natural class of quantum approximate counting problems that captures the complexity of estimating thermal properties of quantum many-body systems. A second consequence is that deciding positivity of Kronecker coefficients is contained in QMA, complementing a recent NP-hardness result of Ikenmeyer, Mulmuley and Walter. We obtain similar results for the related problem of approximating row sums of the character table of the symmetric group. Finally, we discuss an efficient quantum algorithm that approximates normalized Kronecker coefficients to inverse-polynomial additive error.

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arXiv2022

A rapidly mixing Markov chain from any gapped quantum many-body system

We consider the computational task of sampling a bit string $x$ from a distribution $\pi(x)=|\langle x|\psi\rangle|^2$, where $\psi$ is the unique ground state of a local Hamiltonian $H$. Our main result describes a direct link between the inverse spectral gap of $H$ and the mixing time of an associated continuous-time Markov Chain with steady state $\pi$. The Markov Chain can be implemented efficiently whenever ratios of ground state amplitudes $\langle y|\psi\rangle/\langle x|\psi\rangle$ are efficiently computable, the spectral gap of $H$ is at least inverse polynomial in the system size, and the starting state of the chain satisfies a mild technical condition that can be efficiently checked. This extends a previously known relationship between sign-problem free Hamiltonians and Markov chains. The tool which enables this generalization is the so-called fixed-node Hamiltonian construction, previously used in Quantum Monte Carlo simulations to address the fermionic sign problem. We implement the proposed sampling algorithm numerically and use it to sample from the ground state of Haldane-Shastry Hamiltonian with up to 56 qubits. We observe empirically that our Markov chain based on the fixed-node Hamiltonian mixes more rapidly than the standard Metropolis-Hastings Markov chain.

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arXiv2021

How to simulate quantum measurement without computing marginals

We describe and analyze algorithms for classically simulating measurement of an $n$-qubit quantum state $\psi$ in the standard basis, that is, sampling a bit string $x$ from the probability distribution $|\langle x|\psi\rangle|^2$. Our algorithms reduce the sampling task to computing poly$(n)$ amplitudes of $n$-qubit states; unlike previously known techniques they do not require computation of marginal probabilities. First we consider the case where $|\psi\rangle=U|0^n\rangle$ is the output state of an $m$-gate quantum circuit $U$. We propose an exact sampling algorithm which involves computing $O(m)$ amplitudes of $n$-qubit states generated by subcircuits of $U$ spanned by the first $t=1,2,\ldots,m$ gates. We show that our algorithm can significantly accelerate quantum circuit simulations based on tensor network contraction methods or low-rank stabilizer decompositions. As another striking consequence we obtain an efficient classical simulation algorithm for measurement-based quantum computation with the surface code resource state on any planar graph, generalizing a previous algorithm which was known to be efficient only under restrictive topological constraints on the ordering of single-qubit measurements. Second, we consider the case in which $\psi$ is the unique ground state of a local Hamiltonian with a spectral gap that is lower bounded by an inverse polynomial function of $n$. We prove that a simple Metropolis-Hastings Markov Chain mixes rapidly to the desired probability distribution provided that $\psi$ obeys a certain technical condition, which we show is satisfied for all sign-problem free Hamiltonians. This gives a sampling algorithm which involves computing $\mathrm{poly}(n)$ amplitudes of $\psi$.

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arXiv2021

On the complexity of quantum partition functions

The partition function and free energy of a quantum many-body system determine its physical properties in thermal equilibrium. Here we study the computational complexity of approximating these quantities for $n$-qubit local Hamiltonians. First, we report a classical algorithm with $\mathrm{poly}(n)$ runtime which approximates the free energy of a given $2$-local Hamiltonian provided that it satisfies a certain denseness condition. Our algorithm combines the variational characterization of the free energy and convex relaxation methods. It contributes to a body of work on efficient approximation algorithms for dense instances of optimization problems which are hard in the general case, and can be viewed as simultaneously extending existing algorithms for (a) the ground energy of dense $2$-local Hamiltonians, and (b) the free energy of dense classical Ising models. Secondly, we establish polynomial-time equivalence between the problem of approximating the free energy of local Hamiltonians and three other natural quantum approximate counting problems, including the problem of approximating the number of witness states accepted by a QMA verifier. These results suggest that simulation of quantum many-body systems in thermal equilibrium may precisely capture the complexity of a broad family of computational problems that has yet to be defined or characterized in terms of known complexity classes. Finally, we summarize state-of-the-art classical and quantum algorithms for approximating the free energy and show how to improve their runtime and memory footprint.

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arXiv2021

Improved upper bounds on the stabilizer rank of magic states

In this work we improve the runtime of recent classical algorithms for strong simulation of quantum circuits composed of Clifford and T gates. The improvement is obtained by establishing a new upper bound on the stabilizer rank of $m$ copies of the magic state $|T\rangle=\sqrt{2}^{-1}(|0\rangle+e^{i\pi/4}|1\rangle)$ in the limit of large $m$. In particular, we show that $|T\rangle^{\otimes m}$ can be exactly expressed as a superposition of at most $O(2^{\alpha m})$ stabilizer states, where $\alpha\leq 0.3963$, improving on the best previously known bound $\alpha \leq 0.463$. This furnishes, via known techniques, a classical algorithm which approximates output probabilities of an $n$-qubit Clifford + T circuit $U$ with $m$ uses of the T gate to within a given inverse polynomial relative error using a runtime $\mathrm{poly}(n,m)2^{\alpha m}$. We also provide improved upper bounds on the stabilizer rank of symmetric product states $|\psi\rangle^{\otimes m}$ more generally; as a consequence we obtain a strong simulation algorithm for circuits consisting of Clifford gates and $m$ instances of any (fixed) single-qubit $Z$-rotation gate with runtime $\text{poly}(n,m) 2^{m/2}$. We suggest a method to further improve the upper bounds by constructing linear codes with certain properties.

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arXiv2021

Improved approximation algorithms for bounded-degree local Hamiltonians

We consider the task of approximating the ground state energy of two-local quantum Hamiltonians on bounded-degree graphs. Most existing algorithms optimize the energy over the set of product states. Here we describe a family of shallow quantum circuits that can be used to improve the approximation ratio achieved by a given product state. The algorithm takes as input an $n$-qubit product state $|v\rangle$ with mean energy $e_0=\langle v|H|v\rangle$ and variance $\mathrm{Var}=\langle v|(H-e_0)^2|v\rangle$, and outputs a state with an energy that is lower than $e_0$ by an amount proportional to $\mathrm{Var}^2/n$. In a typical case, we have $\mathrm{Var}=\Omega(n)$ and the energy improvement is proportional to the number of edges in the graph. When applied to an initial random product state, we recover and generalize the performance guarantees of known algorithms for bounded-occurrence classical constraint satisfaction problems. We extend our results to $k$-local Hamiltonians and entangled initial states.

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arXiv2021

An area law for 2D frustration-free spin systems

We prove that the entanglement entropy of the ground state of a locally gapped frustration-free 2D lattice spin system satisfies an area law with respect to a vertical bipartition of the lattice into left and right regions. We first establish that the ground state projector of any locally gapped frustration-free 1D spin system can be approximated to within error $\epsilon$ by a degree $O(\sqrt{n\log(\epsilon^{-1})})$ multivariate polynomial in the interaction terms of the Hamiltonian. This generalizes the optimal bound on the approximate degree of the boolean AND function, which corresponds to the special case of commuting Hamiltonian terms. For 2D spin systems we then construct an approximate ground state projector (AGSP) that employs the optimal 1D approximation in the vicinity of the boundary of the bipartition of interest. This AGSP has sufficiently low entanglement and error to establish the area law using a known technique.

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arXiv2021

Classical algorithms for Forrelation

We study the forrelation problem: given a pair of $n$-bit Boolean functions $f$ and $g$, estimate the correlation between $f$ and the Fourier transform of $g$. This problem is known to provide the largest possible quantum speedup in terms of its query complexity and achieves the landmark oracle separation between the complexity class BQP and the Polynomial Hierarchy. Our first result is a classical algorithm for the forrelation problem which has runtime $O(n2^{n/2})$. This is a nearly quadratic improvement over the best previously known algorithm. Secondly, we show that quantum query algorithm that makes $t$ queries to an $n$-bit oracle can be simulated by classical query algorithm making only $O(2^{n(1-1/2t)})$ queries. This fixes a gap in the literature arising from a recently discovered critical error in a previous proof; it matches recently established lower bounds (up to $poly(n,t))$ factors) and thus characterizes the maximal separation in query complexity between quantum and classical algorithms. Finally, we introduce a graph-based forrelation problem where $n$ binary variables live at vertices of some fixed graph and the functions $f,g$ are products of terms describing interactions between nearest-neighbor variables. We show that the graph-based forrelation problem can be solved on a classical computer in time $O(n)$ for any bipartite graph, any planar graph, or, more generally, any graph which can be partitioned into two subgraphs of constant treewidth. The graph-based forrelation is simply related to the variational energy achieved by the Quantum Approximate Optimization Algorithm (QAOA) with two entangling layers and Ising-type cost functions. By exploiting the connection between QAOA and the graph-based forrelation we were able to simulate the recently proposed Recursive QAOA with two entangling layers and $225$ qubits on a laptop computer.

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arXiv2020

Fast simulation of planar Clifford circuits

A general quantum circuit can be simulated classically in exponential time. If it has a planar layout, then a tensor-network contraction algorithm due to Markov and Shi has a runtime exponential in the square root of its size, or more generally exponential in the treewidth of the underlying graph. Separately, Gottesman and Knill showed that if all gates are restricted to be Clifford, then there is a polynomial time simulation. We combine these two ideas and show that treewidth and planarity can be exploited to improve Clifford circuit simulation. Our main result is a classical algorithm with runtime scaling asymptotically as $n^{\omega/2}<n^{1.19}$ which samples from the output distribution obtained by measuring all $n$ qubits of a planar graph state in given Pauli bases. Here $\omega$ is the matrix multiplication exponent. We also provide a classical algorithm with the same asymptotic runtime which samples from the output distribution of any constant-depth Clifford circuit in a planar geometry. Our work improves known classical algorithms with cubic runtime. A key ingredient is a mapping which, given a tree decomposition of some graph $G$, produces a Clifford circuit with a structure that mirrors the tree decomposition and which emulates measurement of the corresponding graph state. We provide a classical simulation of this circuit with the runtime stated above for planar graphs and otherwise $nt^{\omega-1}$ where $t$ is the width of the tree decomposition. Our algorithm incorporates two subroutines which may be of independent interest. The first is a matrix-multiplication-time version of the Gottesman-Knill simulation of multi-qubit measurement on stabilizer states. The second is a new classical algorithm for solving symmetric linear systems over $\mathbb{F}_2$ in a planar geometry, extending previous works which only applied to non-singular linear systems in the analogous setting.

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