Quantum computing and quadratically signed weight enumerators

doi:10.1016/S0020-0190(00)00222-2

Information Processing Letters

Volume 79, Issue 4, 15 August 2001, Pages 173-179

https://doi.org/10.1016/S0020-0190(00)00222-2 Get rights and content

Abstract

We prove that quantum computing is polynomially equivalent to classical probabilistic computing with an oracle for estimating the value of simple sums, quadratically signed weight enumerators (QWGTs). The problem of estimating these sums is cast in terms of promise problems and has two interesting variants. An oracle for the unconstrained variant may be more powerful than quantum computing, while an oracle for a more constrained variant is efficiently solvable in the one-bit model of quantum computing. Thus, problems involving QWGTs yield new problems in BQPP (BQP promise problems) and a natural BQPP complete problem. They can be used to define and study complexity classes and their relationships to quantum computing.

References (19)

P.W. Shor
Algorithms for quantum computation: Discrete logarithms and factoring
P.W. Shor
Polynomial-time algorithms for prime factorization and discrete logarithms on a quantum computer
SIAM J. Comput.
(1997)
P.W. Shor
Fault-tolerant quantum computation
A.Y. Kitaev
Quantum computations: algorithms and error correction
Uspekhi Mat. Nauk
(1997)
E. Knill et al.
Resilient quantum computation
Science
(1998)
D. Aharonov et al.
Fault-tolerant quantum computation with constant error
D. Aharonov et al.
Fault-tolerant quantum computation with constant error
E. Bernstein et al.
Quantum complexity theory
SIAM J. Comput.
(1997)
E. Knill et al.
On the power of one bit of quantum information
Phys. Rev. Lett.
(1998)

There are more references available in the full text version of this article.

Cited by (29)

Sumcheck-based delegation of quantum computing to rational server
2022, Theoretical Computer Science
Delegated quantum computing enables a client with weak computational power to delegate quantum computing to a remote quantum server in such a way that the integrity of the server can be efficiently verified by the client. Recently, a new model of delegated quantum computing has been proposed, namely, rational delegated quantum computing. In this model, after the client interacts with the server, the client pays a reward to the server depending on the server's messages and the client's random bits. The rational server sends messages that maximize the expected value of the reward. It is known that the classical client can delegate universal quantum computing to the rational quantum server in one round. In this paper, we propose novel one-round rational delegated quantum computing protocols by generalizing the classical rational sumcheck protocol. An advantage of our protocols is that they are gate-set independent: the construction of the previous rational protocols depends on gate sets, while our sumcheck technique can be easily realized with any local gate set (each of whose elementary gates can be specified with a polynomial number of bits). Furthermore, as with the previous protocols, our reward function satisfies natural requirements (the reward is non-negative, upper-bounded by a constant, and its maximum expected value is lower-bounded by a constant). We also discuss the reward gap. Simply speaking, the reward gap is a minimum loss on the expected value of the server's reward incurred by the server's behavior that makes the client accept an incorrect answer. The reward gap should therefore be large enough to incentivize the server to behave optimally. Although our sumcheck-based protocols have only exponentially small reward gaps as in the previous protocols, we show that a constant reward gap can be achieved if two noncommunicating but entangled rational servers are allowed. We also discuss whether a single rational server is sufficient under the (widely believed) assumption that the learning-with-errors problem is hard for polynomial-time quantum computing. Apart from these results, we show, under a certain condition, the equivalence between rational and ordinary delegated quantum computing protocols. This equivalence then serves as a basis for a reward-gap amplification method.
Quasi-probability representations of quantum theory with applications to quantum information science
2011, Reports on Progress in Physics
Quantum discord and quantum computing - An appraisal
2011, International Journal of Quantum Information
On the quantum computational complexity of the Ising spin glass partition function and of knot invariants
2004, New Journal of Physics
Estimating distinguishability measures on quantum computers
2023, Physical Review A
Estimating distinguishability measures on quantum computers
2021, arXiv

View all citing articles on Scopus

View full text

Quantum computing and quadratically signed weight enumerators

Abstract

Algorithms for quantum computation: Discrete logarithms and factoring

Polynomial-time algorithms for prime factorization and discrete logarithms on a quantum computer

SIAM J. Comput.

Fault-tolerant quantum computation

Quantum computations: algorithms and error correction

Uspekhi Mat. Nauk

Resilient quantum computation

Science

Fault-tolerant quantum computation with constant error

Fault-tolerant quantum computation with constant error

Quantum complexity theory

SIAM J. Comput.

On the power of one bit of quantum information

Phys. Rev. Lett.