61 research outputs found
How Hard Is Deciding Trivial Versus Nontrivial in the Dihedral Coset Problem?
We study the hardness of the dihedral hidden subgroup problem. It is known that lattice problems reduce to it, and that it reduces to random subset sum with density > 1 and also to quantum sampling subset sum solutions. We examine a decision version of the problem where the question asks whether the hidden subgroup is trivial or order two. The decision problem essentially asks if a given vector is in the span of all coset states. We approach this by first computing an explicit basis for the coset space and the perpendicular space. We then look at the consequences of having efficient unitaries that use this basis. We show that if a unitary maps the basis to the standard basis in any way, then that unitary can be used to solve random subset sum with constant density >1. We also show that if a unitary can exactly decide membership in the coset subspace, then the collision problem for subset sum can be solved for density >1 but approaching 1 as the problem size increases. This strengthens the previous hardness result that implementing the optimal POVM in a specific way is as hard as quantum sampling subset sum solutions
Efficient learning of -doped stabilizer states with single-copy measurements
One of the primary objectives in the field of quantum state learning is to
develop algorithms that are time-efficient for learning states generated from
quantum circuits. Earlier investigations have demonstrated time-efficient
algorithms for states generated from Clifford circuits with at most
non-Clifford gates. However, these algorithms necessitate multi-copy
measurements, posing implementation challenges in the near term due to the
requisite quantum memory. On the contrary, using solely single-qubit
measurements in the computational basis is insufficient in learning even the
output distribution of a Clifford circuit with one additional gate under
reasonable post-quantum cryptographic assumptions. In this work, we introduce
an efficient quantum algorithm that employs only nonadaptive single-copy
measurement to learn states produced by Clifford circuits with a maximum of
non-Clifford gates, filling a gap between the previous positive and
negative results.Comment: 6 page
Non-Interactive Classical Verification of Quantum Depth: A Fine-Grained Characterization
We introduce protocols for classical verification of quantum depth (CVQD). These protocols enable a classical verifier to differentiate between devices of varying quantum circuit depths, even in the presence of classical computation. The goal is to demonstrate that a classical verifier can reject a device with a quantum circuit depth of no more than , even if the prover employs additional polynomial-time classical computation to deceive. Conversely, the verifier accepts a device with a quantum circuit depth of .
Previous results for separating hybrid quantum-classical computers with various quantum depths require either quantum access to oracles or interactions between the classical verifier and the quantum prover. However, instantiating oracle separations can significantly increase the quantum depth in general, and interaction challenges the quantum device to keep the qubits coherent while waiting for the verifier\u27s messages. These requirements pose barriers to implementing the protocols on near-term devices.
In this work, we present a two-message protocol under the quantum hardness of learning with errors and the random oracle heuristic. An honest prover only needs classical access to the random oracle, and therefore any instantiation of the oracle does not increase the quantum depth. To our knowledge, our protocol is the first non-interactive CVQD, the instantiation of which using concrete hash functions, e.g., SHA-3, does not require additional quantum depth.
Our second protocol seeks to explore the minimality of cryptographic assumptions and the tightness of the separations. To accomplish this, we introduce an untrusted quantum machine that shares entanglements with the target machine. Utilizing a robust self-test, our protocol certifies the depth of the target machine with information-theoretic security and nearly optimal separation
Quantum Meets the Minimum Circuit Size Problem
In this work, we initiate the study of the Minimum Circuit Size Problem (MCSP) in the quantum setting. MCSP is a problem to compute the circuit complexity of Boolean functions. It is a fascinating problem in complexity theory - its hardness is mysterious, and a better understanding of its hardness can have surprising implications to many fields in computer science.
We first define and investigate the basic complexity-theoretic properties of minimum quantum circuit size problems for three natural objects: Boolean functions, unitaries, and quantum states. We show that these problems are not trivially in NP but in QCMA (or have QCMA protocols). Next, we explore the relations between the three quantum MCSPs and their variants. We discover that some reductions that are not known for classical MCSP exist for quantum MCSPs for unitaries and states, e.g., search-to-decision reductions and self-reductions. Finally, we systematically generalize results known for classical MCSP to the quantum setting (including quantum cryptography, quantum learning theory, quantum circuit lower bounds, and quantum fine-grained complexity) and also find new connections to tomography and quantum gravity. Due to the fundamental differences between classical and quantum circuits, most of our results require extra care and reveal properties and phenomena unique to the quantum setting. Our findings could be of interest for future studies, and we post several open problems for further exploration along this direction
Quantum-Inspired Sublinear Algorithm for Solving Low-Rank Semidefinite Programming
Semidefinite programming (SDP) is a central topic in mathematical
optimization with extensive studies on its efficient solvers. In this paper, we
present a proof-of-principle sublinear-time algorithm for solving SDPs with
low-rank constraints; specifically, given an SDP with constraint matrices,
each of dimension and rank , our algorithm can compute any entry and
efficient descriptions of the spectral decomposition of the solution matrix.
The algorithm runs in time
given access to a sampling-based low-overhead data structure for the constraint
matrices, where is the precision of the solution. In addition, we
apply our algorithm to a quantum state learning task as an application.
Technically, our approach aligns with 1) SDP solvers based on the matrix
multiplicative weight (MMW) framework by Arora and Kale [TOC '12]; 2)
sampling-based dequantizing framework pioneered by Tang [STOC '19]. In order to
compute the matrix exponential required in the MMW framework, we introduce two
new techniques that may be of independent interest:
Weighted sampling: assuming sampling access to each individual
constraint matrix , we propose a procedure that gives a
good approximation of .
Symmetric approximation: we propose a sampling procedure that gives
the \emph{spectral decomposition} of a low-rank Hermitian matrix . To the
best of our knowledge, this is the first sampling-based algorithm for spectral
decomposition, as previous works only give singular values and vectors.Comment: 37 pages, 1 figure. To appear in the Proceedings of the 45th
International Symposium on Mathematical Foundations of Computer Science (MFCS
2020
On the Impossibility of Post-Quantum Black-Box Zero-Knowledge in Constant Rounds
We investigate the existence of constant-round post-quantum black-box
zero-knowledge protocols for . As a main result, we show that
there is no constant-round post-quantum black-box zero-knowledge argument for
unless . As constant-round
black-box zero-knowledge arguments for exist in the classical
setting, our main result points out a fundamental difference between
post-quantum and classical zero-knowledge protocols. Combining previous
results, we conclude that unless ,
constant-round post-quantum zero-knowledge protocols for exist if
and only if we use non-black-box techniques or relax certain security
requirements such as relaxing standard zero-knowledge to
-zero-knowledge. Additionally, we also prove that three-round and
public-coin constant-round post-quantum black-box -zero-knowledge
arguments for do not exist unless .Comment: 46 page
MUJER EN EL BOSQUE CON UN PERRO A SUS PIES [Material gráfico]
Copia digital. Madrid : Ministerio de Educación, Cultura y Deporte, 201
Quantum-Inspired Algorithms for Solving Low-Rank Linear Equation Systems with Logarithmic Dependence on the Dimension
We present two efficient classical analogues of the quantum matrix inversion algorithm [16] for low-rank matrices. Inspired by recent work of Tang [27], assuming length-square sampling access to input data, we implement the pseudoinverse of a low-rank matrix allowing us to sample from the solution to the problem Ax = b using fast sampling techniques. We construct implicit descriptions of the pseudo-inverse by finding approximate singular value decomposition of A via subsampling, then inverting the singular values. In principle, our approaches can also be used to apply any desired “smooth” function to the singular values. Since many quantum algorithms can be expressed as a singular value transformation problem [15], our results indicate that more low-rank quantum algorithms can be effectively “dequantised” into classical length-square sampling algorithms
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