Quantum Algorithms for Some Hidden Shift Problems

Dam, Wim van; Hallgren, Sean; Ip, Lawrence

doi:10.1137/s009753970343141x

Cited by 133 publications

(141 citation statements)

References 30 publications

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“…Specifically, we simulated a quantum algorithm which solves the hidden shift problem [21] for non-linear Boolean functions [22]. An instance of the hidden shift problem is defined by a pair of oracle functions f, f 0 ∶F n 2 → fAE1g and a hidden shift string s ∈ F n 2 .…”

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confidence: 99%

Improved Classical Simulation of Quantum Circuits Dominated by Clifford Gates

Bravyi¹,

2016

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We present a new algorithm for classical simulation of quantum circuits over the Clifford þ T gate set. The runtime of the algorithm is polynomial in the number of qubits and the number of Clifford gates in the circuit but exponential in the number of T gates. The exponential scaling is sufficiently mild that the algorithm can be used in practice to simulate medium-sized quantum circuits dominated by Clifford gates. The first demonstrations of fault-tolerant quantum circuits based on 2D topological codes are likely to be dominated by Clifford gates due to a high implementation cost associated with logical T gates. Thus our algorithm may serve as a verification tool for near-term quantum computers which cannot in practice be simulated by other means. To demonstrate the power of the new method, we performed a classical simulation of a hidden shift quantum algorithm with 40 qubits, a few hundred Clifford gates, and nearly 50 T gates. DOI: 10.1103/PhysRevLett.116.250501 The path towards building a large-scale quantum computer will inevitably require verification and validation of small quantum devices. One way to check that such a device is working properly is to simulate it on a classical computer. This becomes impractical at some point because the cost of classical simulation typically grows exponentially with the size of a quantum system. With this fundamental limitation in mind it is natural to ask how well we can do in practice.Simulation methods which store a complete description of an n-qubit quantum state as a complex vector of size 2 n are limited to a small number of qubits n ≈ 30 − 40. For example, a state-of-the art implementation has been used to simulate Shor's factoring algorithm with 31 qubits and roughly half a million gates [1]. Using distributed computation it is possible to simulate 40 qubit circuits [2]. For certain restricted classes of quantum circuits it is possible to do much better [3][4][5][6][7]. Most significantly, the GottesmanKnill theorem allows efficient classical simulation of quantum circuits composed of gates in the so-called Clifford group [3]. In practice this allows one to simulate such circuits with thousands of qubits [1,4]. It also means that a quantum computer will need to use gates outside of the Clifford group in order to achieve useful speedups over classical computation. The full power of quantum computation can be recovered by adding a single non-Clifford gate to the Clifford group. A simple choice is the single-qubit T ¼ j0ih0j þ e iπ=4 j1ih1j gate. The Clifford þ T gate set obtained in this way is a natural instruction set for smallscale fault-tolerant quantum computers based on the surface code [8,9], and has been at the center of a recent renaissance in classical techniques for compiling quantum circuits [10][11][12].When it comes to realizing a logical (encoded) circuit, non-Clifford gates pose a serious challenge for any faulttolerant scheme based on 2D stabilizer codes [8,13] due to the lack of topological protection [14,15]. Such nonClifford gates can be imp...

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confidence: 99%

Improved Classical Simulation of Quantum Circuits Dominated by Clifford Gates

Bravyi¹,

2016

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“…We believe that general SHSP is actually much harder than graph isomorphism. If graph isomorphism does admit a special quantum algorithm, it could be analogous to a quantum polynomial time algorithm found by Van Dam, Hallgren, and Ip [24] for certain special abelian hidden shift problems. (In particular their algorithm applies to the Legendre symbol with a hidden shift.)…”

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confidence: 99%

A Subexponential-Time Quantum Algorithm for the Dihedral Hidden Subgroup Problem

Kuperberg¹

2005

SIAM J. Comput.

274

254

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We present a quantum algorithm for the dihedral hidden subgroup problem with time and query complexity 2 O( √ log N) . In this problem an oracle computes a function f on the dihedral group D N which is invariant under a hidden reflection in D N . By contrast the classical query complexity of DHSP is O( √ N). The algorithm also applies to the hidden shift problem for an arbitrary finitely generated abelian group.The algorithm begins as usual with a quantum character transform, which in the case of D N is essentially the abelian quantum Fourier transform. This yields the name of a group representation of D N , which is not by itself useful, and a state in the representation, which is a valuable but indecipherable qubit. The algorithm proceeds by repeatedly pairing two unfavorable qubits to make a new qubit in a more favorable representation of D N . Once the algorithm obtains certain target representations, direct measurements reveal the hidden subgroup.

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“…In this section we consider a PRF based on the Legendre symbol, which to the best of our knowledge was first described in [44]. Whilst this PRF is very inefficient when applied to cleartext data, we show that with secret-shared data in the MPC setting it allows for a very simple protocol.…”

Section: Prf From the Legendre Symbolmentioning

confidence: 99%

“…Van Dam, Hallgren and Ip [44] described a quantum polynomial time algorithm for the SLS problem that recovers the secret k if the oracle can be queried on a quantum state. They conjectured that classically, there is no polynomial time algorithm for this problem.…”

Section: Hardness Of the Shifted Legendre Symbol Problemmentioning

confidence: 99%

MPC-Friendly Symmetric Key Primitives

Grassi

Rechberger

Rotaru

et al. 2016

Proceedings of the 2016 ACM SIGSAC Conference on Computer and Communications Security

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We discuss the design of symmetric primitives, in particular Pseudo-Random Functions (PRFs) which are suitable for use in a secret-sharing based MPC system. We consider three different PRFs: the Naor-Reingold PRF, a PRF based on the Legendre symbol, and a specialized block cipher design called MiMC. We present protocols for implementing these PRFs within a secret-sharing based MPC system, and discuss possible applications. We then compare the performance of our protocols. Depending on the application, different PRFs may offer different optimizations and advantages over the classic AES benchmark. Thus, we cannot conclude that there is one optimal PRF to be used in all situations.

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Quantum Algorithms for Some Hidden Shift Problems

Cited by 133 publications

References 30 publications

Improved Classical Simulation of Quantum Circuits Dominated by Clifford Gates

Improved Classical Simulation of Quantum Circuits Dominated by Clifford Gates

A Subexponential-Time Quantum Algorithm for the Dihedral Hidden Subgroup Problem

MPC-Friendly Symmetric Key Primitives

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