arXiv subjects

Anurag Anshu

Publications and source records attributed to Anurag Anshu.

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.

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arXiv2026

Classical Adversarial Fault-Tolerance and PCPs

We show how to compile an arbitrary classical circuit into a fault-tolerant circuit, which performs the desired computation even when an almost-linear number of bits are adversarially chosen and corrupted in each timestep. Using a variant of this fault-tolerance scheme that only detects (rather than corrects) corruptions, we give a new construction of probabilistically checkable proofs (PCPs) for NP with polylogarithmic query complexity. This PCP construction from fault-tolerance presents a promising candidate for quantization by the work of Anshu, Breuckmann, and Nguyen (STOC'24), who provided a roadmap for constructing quantum PCPs via fault-tolerance.

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arXiv2026

An energy-based uncertainty principle and low-energy state preparation

Preparing low-energy states of many-body Hamiltonians is a central challenge in quantum computing, quantum complexity, and condensed matter physics. Existing approaches often get trapped in suboptimal states such as high-energy eigenstates or, more generally, low-variance states that resist further energy reduction. In this work, we explore a different perspective: instead of optimizing with respect to a single Hamiltonian, we leverage the fact that many systems admit families of Hamiltonians that share similar low-energy subspaces but differ at higher energies. We show that this redundancy can be turned into an algorithmic resource by establishing an energy-based uncertainty principle, which implies that these Hamiltonians cannot simultaneously admit low-variance states at higher energies. This suggests a simple strategy of alternating energy-lowering steps across such Hamiltonians, which we investigate numerically on several models. We also introduce a sparse variant where the uncertainty principle yields quadratically larger variance at higher energies, leading to more pronounced energy change. Overall, this work suggests a range of open questions at the interface of random matrix theory, local Hamiltonians and low-energy state preparation, aimed at understanding when such approaches are practical and how they can be analyzed rigorously.

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arXiv2026

Randomness compression in communication networks

Given a correlation generated by a (possibly quantum) communication network, we study the amount of shared randomness required to generate it. We develop a novel upper bound for approximating distributions generated by arbitrary networks and showcase instances where it significantly outperforms the best-known upper bounds for the exact case. This demonstrates that one can have substantial savings in resources if small perturbations are acceptable. We derive our bound using Hoeffding's inequality and apply it to various commonly-used communication networks such as the Bell scenario and triangle scenario.

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arXiv2025

Learning quantum Gibbs states locally and efficiently

Learning the Hamiltonian underlying a quantum many-body system in thermal equilibrium is a fundamental task in quantum learning theory and experimental sciences. To learn the Gibbs state of local Hamiltonians at any inverse temperature $\beta$, the state-of-the-art provable algorithms fall short of the optimal sample and computational complexity, in sharp contrast with the locality and simplicity in the classical cases. In this work, we present a learning algorithm that learns each local term of a $n$-qubit $D$-dimensional Hamiltonian to an additive error $\epsilon$ with sample complexity $\tilde{O}\left(\frac{e^{\mathrm{poly}(\beta)}}{\beta^2\epsilon^2}\right)\log(n)$. The protocol uses parallelizable local quantum measurements that act within bounded regions of the lattice and near-linear-time classical post-processing. Thus, our complexity is near optimal with respect to $n,\epsilon$ and is polynomially tight with respect to $\beta$. We also give a learning algorithm for Hamiltonians with bounded interaction degree with sample and time complexities of similar scaling on $n$ but worse on $\beta, \epsilon$. At the heart of our algorithm is the interplay between locality, the Kubo-Martin-Schwinger condition, and the operator Fourier transform at arbitrary temperatures.

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arXiv2024

On the complexity of unique quantum witnesses and quantum approximate counting

We study the long-standing open question on the power of unique witnesses in quantum protocols, which asks if $\textsf{UniqueQMA}$, a variant of $\textsf{QMA}$ whose accepting witness space is 1-dimensional, contains $\mathsf{QMA}$ under quantum reductions. This work rules out any black-box reduction from $\mathsf{QMA}$ to $\mathsf{UniqueQMA}$ by showing a quantum oracle separation between $\mathsf{BQP}^\mathsf{UniqueQMA}$ and $\mathsf{QMA}$. This provides a contrast to the classical case, where the Valiant-Vazirani theorem shows a black-box randomized reduction from $\mathsf{UniqueNP}$ to $\mathsf{NP}$, and suggests the need for studying the structure of the ground space of local Hamiltonians in distilling a potential unique witness. Via similar techniques, we show, relative to a quantum oracle, that $\mathsf{QMA}^\mathsf{QMA}$ cannot decide quantum approximate counting, ruling out a quantum analogue of Stockmeyer's algorithm in the black-box setting. We then ask a natural question; what structural properties of the local Hamiltonian problem can we exploit? We introduce a physically motivated candidate by showing that the ground energy of local Hamiltonians that satisfy a computational variant of the eigenstate thermalization hypothesis (ETH) can be estimated through a $\mathsf{UniqueQMA}$ protocol. Our protocol can be viewed as a quantum expander test in a low energy subspace of the Hamiltonian and verifies a unique entangled state across two copies of the subspace. This allows us to conclude that if $\mathsf{UniqueQMA}$ is not equivalent to $\mathsf{QMA}$, then $\mathsf{QMA}$-hard Hamiltonians must violate ETH under adversarial perturbations. This also serves as evidence that chaotic local Hamiltonians, such as the SYK model may be computationally simpler than general local Hamiltonians.

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arXiv2024

On the Computational Power of QAC0 with Barely Superlinear Ancillae

$\mathrm{QAC}^0$ is the family of constant-depth polynomial-size quantum circuits consisting of arbitrary single qubit unitaries and multi-qubit Toffoli gates. It was introduced by Moore [arXiv: 9903046] as a quantum counterpart of $\mathrm{AC}^0$, along with the conjecture that $\mathrm{QAC}^0$ circuits can not compute PARITY. In this work we make progress on this longstanding conjecture: we show that any depth-$d$ $\mathrm{QAC}^0$ circuit requires $n^{1+3^{-d}}$ ancillae to compute a function with approximate degree $\Theta(n)$, which includes PARITY, MAJORITY and $\mathrm{MOD}_k$. We further establish superlinear lower bounds on quantum state synthesis and quantum channel synthesis. This is the first superlinear lower bound on the super-linear sized $\mathrm{QAC}^0$. Regarding PARITY, we show that any further improvement on the size of ancillae to $n^{1+\exp(-o(d))}$ would imply that PARITY $\not\in$ QAC0. These lower bounds are derived by giving low-degree approximations to $\mathrm{QAC}^0$ circuits. We show that a depth-$d$ $\mathrm{QAC}^0$ circuit with $a$ ancillae, when applied to low-degree operators, has a degree $(n+a)^{1-3^{-d}}$ polynomial approximation in the spectral norm. This implies that the class $\mathrm{QLC}^0$, corresponding to linear size $\mathrm{QAC}^0$ circuits, has approximate degree $o(n)$. This is a quantum generalization of the result that $\mathrm{LC}^0$ circuits have approximate degree $o(n)$ by Bun, Robin, and Thaler [SODA 2019]. Our result also implies that $\mathrm{QLC}^0\neq\mathrm{NC}^1$.

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arXiv2024

Learning shallow quantum circuits

Despite fundamental interests in learning quantum circuits, the existence of a computationally efficient algorithm for learning shallow quantum circuits remains an open question. Because shallow quantum circuits can generate distributions that are classically hard to sample from, existing learning algorithms do not apply. In this work, we present a polynomial-time classical algorithm for learning the description of any unknown $n$-qubit shallow quantum circuit $U$ (with arbitrary unknown architecture) within a small diamond distance using single-qubit measurement data on the output states of $U$. We also provide a polynomial-time classical algorithm for learning the description of any unknown $n$-qubit state $\lvert \psi \rangle = U \lvert 0^n \rangle$ prepared by a shallow quantum circuit $U$ (on a 2D lattice) within a small trace distance using single-qubit measurements on copies of $\lvert \psi \rangle$. Our approach uses a quantum circuit representation based on local inversions and a technique to combine these inversions. This circuit representation yields an optimization landscape that can be efficiently navigated and enables efficient learning of quantum circuits that are classically hard to simulate.

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arXiv2023

Circuit-to-Hamiltonian from tensor networks and fault tolerance

We define a map from an arbitrary quantum circuit to a local Hamiltonian whose ground state encodes the quantum computation. All previous maps relied on the Feynman-Kitaev construction, which introduces an ancillary `clock register' to track the computational steps. Our construction, on the other hand, relies on injective tensor networks with associated parent Hamiltonians, avoiding the introduction of a clock register. This comes at the cost of the ground state containing only a noisy version of the quantum computation, with independent stochastic noise. We can remedy this - making our construction robust - by using quantum fault tolerance. In addition to the stochastic noise, we show that any state with energy density exponentially small in the circuit depth encodes a noisy version of the quantum computation with adversarial noise. We also show that any `combinatorial state' with energy density polynomially small in depth encodes the quantum computation with adversarial noise. This serves as evidence that any state with energy density polynomially small in depth has a similar property. As applications, we give a new proof of the QMA-completeness of the local Hamiltonian problem (with logarithmic locality) and show that contracting injective tensor networks to additive error is BQP-hard. We also discuss the implication of our construction to the quantum PCP conjecture, combining with an observation that QMA verification can be done in logarithmic depth.

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arXiv2023

Fermionic Hamiltonians without trivial low-energy states

We construct local fermionic Hamiltonians with no low-energy trivial states (NLTS), providing a fermionic counterpart to the NLTS theorem. Distinctly from the qubit case, we define trivial states via finite-depth $\textit{fermionic}$ quantum circuits. We furthermore allow free access to Gaussian fermionic operations, provided they involve at most $O(n)$ ancillary fermions. The desired fermionic Hamiltonian can be constructed using any qubit Hamiltonian which itself has the NLTS property via well-spread distributions over bitstrings, such as the construction in [Anshu, Breuckmann, Nirkhe, STOC 2023]. We define a fermionic analogue of the class quantum PCP and discuss its relation with the qubit version.

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arXiv2023

A survey on the complexity of learning quantum states

We survey various recent results that rigorously study the complexity of learning quantum states. These include progress on quantum tomography, learning physical quantum states, alternate learning models to tomography and learning classical functions encoded as quantum states. We highlight how these results are paving the way for a highly successful theory with a range of exciting open questions. To this end, we distill 25 open questions from these results.

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arXiv2022

Concentration bounds for quantum states and limitations on the QAOA from polynomial approximations

We prove concentration bounds for the following classes of quantum states: (i) output states of shallow quantum circuits, answering an open question from [DPMRF22]; (ii) injective matrix product states; (iii) output states of dense Hamiltonian evolution, i.e. states of the form $e^{\iota H^{(p)}} \cdots e^{\iota H^{(1)}} |\psi_0\rangle$ for any $n$-qubit product state $|\psi_0\rangle$, where each $H^{(i)}$ can be any local commuting Hamiltonian satisfying a norm constraint, including dense Hamiltonians with interactions between any qubits. Our proofs use polynomial approximations to show that these states are close to local operators. This implies that the distribution of the Hamming weight of a computational basis measurement (and of other related observables) concentrates. An example of (iii) are the states produced by the quantum approximate optimisation algorithm (QAOA). Using our concentration results for these states, we show that for a random spin model, the QAOA can only succeed with negligible probability even at super-constant level $p = o(\log \log n)$, assuming a strengthened version of the so-called overlap gap property. This gives the first limitations on the QAOA on dense instances at super-constant level, improving upon the recent result [BGMZ22].

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arXiv2022

NLTS Hamiltonians from good quantum codes

The NLTS (No Low-Energy Trivial State) conjecture of Freedman and Hastings [2014] posits that there exist families of Hamiltonians with all low energy states of non-trivial complexity (with complexity measured by the quantum circuit depth preparing the state). We prove this conjecture by showing that the recently discovered families of constant-rate and linear-distance QLDPC codes correspond to NLTS local Hamiltonians.

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arXiv2022

A construction of Combinatorial NLTS

The NLTS (No Low-Energy Trivial State) conjecture of Freedman and Hastings [2014] posits that there exist families of Hamiltonians with all low energy states of high complexity (with complexity measured by the quantum circuit depth preparing the state). Here, we prove a weaker version called the combinatorial NLTS, where a quantum circuit lower bound is shown against states that violate a (small) constant fraction of local terms. This generalizes the prior NLETS results (Eldar and Harrow [2017]; Nirkhe, Vazirani and Yuen [2018]). Our construction is obtained by combining tensor networks with expander codes (Sipser and Spielman [1996]). The Hamiltonian is the parent Hamiltonian of a perturbed tensor network, inspired by the `uncle Hamiltonian' of Fernandez-Gonzalez et. al. [2015]. Thus, we deviate from the quantum CSS code Hamiltonians considered in most prior works.

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arXiv2021

Distributed quantum inner product estimation

As small quantum computers are becoming available on different physical platforms, a benchmarking task known as cross-platform verification has been proposed that aims to estimate the fidelity of states prepared on two quantum computers. This task is fundamentally distributed, as no quantum communication can be performed between the two physical platforms due to hardware constraints, which prohibits a joint SWAP test. In this paper we settle the sample complexity of this task across all measurement and communication settings. The essence of the task, which we call distributed quantum inner product estimation, involves two players Alice and Bob who have $k$ copies of unknown states $\rho,\sigma$ (acting on $\mathbb{C}^{d}$) respectively. Their goal is to estimate $\mathrm{Tr}(\rho\sigma)$ up to additive error $\varepsilon\in(0,1)$, using local quantum operations and classical communication. In the weakest setting where only non-adaptive single-copy measurements and simultaneous message passing are allowed, we show that $k=O(\max\{1/\varepsilon^2,\sqrt{d}/\varepsilon\})$ copies suffice. This achieves a savings compared to full tomography which takes $\Omega(d^3)$ copies with single-copy measurements. Surprisingly, we also show that the sample complexity must be at least $\Omega(\max\{1/\varepsilon^2,\sqrt{d}/\varepsilon\})$, even in the strongest setting where adaptive multi-copy measurements and arbitrary rounds of communication are allowed. This shows that the success achieved by shadow tomography, for sample-efficiently learning the properties of a single system, cannot be generalized to the distributed setting. Furthermore, the fact that the sample complexity remains the same with single and multi-copy measurements contrasts with single system quantum property testing, which often demonstrate exponential separations in sample complexity with single and multi-copy measurements.

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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

One-shot quantum state redistribution and quantum Markov chains

We revisit the task of quantum state redistribution in the one-shot setting, and design a protocol for this task with communication cost in terms of a measure of distance from quantum Markov chains. More precisely, the distance is defined in terms of quantum max-relative entropy and quantum hypothesis testing entropy. Our result is the first to operationally connect quantum state redistribution and quantum Markov chains, and can be interpreted as an operational interpretation for a possible one-shot analogue of quantum conditional mutual information. The communication cost of our protocol is lower than all previously known ones and asymptotically achieves the well-known rate of quantum conditional mutual information. Thus, our work takes a step towards the important open question of near-optimal characterization of the one-shot quantum state redistribution.

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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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