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 ...
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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.
This paper introduces a novel abstraction for programming quantum operations, specifically projective Cliffords, as functions over the qudit Pauli group. We define a categorical semantics for projective Cliffords base...
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We provide a careful analysis of the structure theorem for the n-qudit projective Clifford group and various encoding schemes for its elements. In particular, we derive formulas for evaluation, composition, and invers...
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How many T gates are needed to approximate an arbitrary n-qubit quantum state to within error Ε? Improving prior work of Low, Kliuchnikov, and Schaeffer, we show that the optimal asymptotic scaling is Θ (Formula Pre...
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We establish the conditions for ‘equivalent neural networks’ - neural networks with different weights, biases, and threshold functions which result in the same associated function. We prove that given a neural netwo...
We establish the conditions for ‘equivalent neural networks’ - neural networks with different weights, biases, and threshold functions which result in the same associated function. We prove that given a neural network $\mathcal{N}$ with piecewise linear activation, the space of coefficients describing all equivalent neural networks is given by a semialgebraic set. This result is obtained by studying different representations of a given piecewise linear function using the Tarski-Seidenberg theorem. Given a neural architecture and an initial neural network $\mathcal{N}_{0}$ , specified by coefficients, we give an algorithm to compute inequalities defining the corresponding semiaglebraic sets. These algorithms are based on the rules of max-plus algebra.
We investigate the problem of efficiently estimating expectation values of large sets of observables from copies of an unknown many-body quantum state. This task, known as shadow tomography, is crucial for quantum sim...
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We investigate the problem of efficiently estimating expectation values of large sets of observables from copies of an unknown many-body quantum state. This task, known as shadow tomography, is crucial for quantum simulation and quantum chemistry, where the relevant observables often include k-body fermionic or k-local Pauli operators. A key goal is to achieve sample efficiency, computational efficiency, and few-copy measurements. We introduce the notion of triply efficient shadow tomography to formalize these requirements. Prior work has shown that single-copy measurements suffice for k-local Pauli operators with constant k. However, we prove that for k-body fermionic observables and the full set of n-qubit Pauli operators, single-copy protocols fail to achieve sample-efficient tomography. We then present new protocols based on measuring two copies of the state at a time that overcome these lower bounds, providing a triply efficient protocol.
We give a new definition of self-testing for correlations in terms of states on C∗-algebras. We show that this definition is equivalent to the standard definition for any class of finite-dimensional quantum models whi...
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In this paper we consider the compression of asymptotically many i.i.d. copies of ensembles of mixed quantum states where the encoder has access to a side information system. This source is equivalently defined as a c...
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quantum nondeterministic distributed computing was recently introduced as dQMA (distributed quantum Merlin-Arthur) protocols by Fraigniaud, Le Gall, Nishimura and Paz (ITCS 2021). In dQMA protocols, with the help of q...
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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 measu...
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