is the class of problems solvable in quantum polynomial time, given a polynomial-size “quantum advice state” that depends only on the input length. This resolves a question of Buhrman, and means that we should not hope for an unrelativized separation between quantum and classical advice. Underlying our complexity result is a general new relation between deterministic and quantum one-way communication complexities, which applies to partial as well as total functions. Second, we construct an oracle relative to which

\mathsf{NP}\not \subset \mathsf{BQP/qpoly}

. To do so, we use the polynomial method to give the first correct proof of a direct product theorem for quantum search. This theorem has other applications; for example, it can be used to fix a result of Klauck about quantum time-space tradeoffs for sorting. Third, we introduce a new trace distance method for proving lower bounds on quantum one-way communication complexity. Using this method, we obtain optimal quantum lower bounds for two problems of Ambainis, for which no nontrivial lower bounds were previously known even for classical randomized protocols. A preliminary version of this paper appeared in the 2004 Conference on Computational Complexity (CCC).

DOI: 10.4086/toc.2005.v001a001

Related papers

Limitations of quantum advice and one-way communication

Scott Aaronson

Although a quantum state requires exponentially many classical bits to describe, the laws of quantum mechanics impose severe restrictions on how that state can be accessed. This paper shows in three settings that quantum messages have only limited advantages over classical ones. First, we show that BQP/qpoly /spl sube/ PP/poly, where BQP/qpoly is the class of problems solvable in quantum polynomial time, given a polynomial-size "quantum advice state " that depends only on the input length. This resolves a question of Buhrman, and means that we should not hope for an unrelativized separation between quantum and classical advice. Underlying our complexity result is a general relation between deterministic and quantum one-way communication complexities, which applies to partial as well as total functions. Second, we construct an oracle relative to which NP /spl nsub/ BQP/qpoly. To do so, we use the polynomial method to give the first correct proof of a direct product theorem for quantum search. This theorem has many other applications; for example, it can be used to fix and even improve on a flawed result of Klauck about quantum time-space tradeoffs for sorting. Third, we introduce a trace distance method for proving lower bounds on quantum one-way communication complexity. Using this method, we obtain optimal quantum lower bounds for two problems of Ambainis, for which no nontrivial lower bounds were previously known even for classical randomized protocols.Although a quantum state requires exponentially many classical bits to describe, the laws of quantum mechanics impose severe restrictions on how that state can be accessed. This paper shows in three settings that quantum messages have only limited advantages over classical ones. First, we show that BQP/qpoly /spl sube/ PP/poly, where BQP/qpoly is the class of problems solvable in quantum polynomial time, given a polynomial-size "quantum advice state " that depends only on the input length. This resolves a question of Buhrman, and means that we should not hope for an unrelativized separation between quantum and classical advice. Underlying our complexity result is a general relation between deterministic and quantum one-way communication complexities, which applies to partial as well as total functions. Second, we construct an oracle relative to which NP /spl nsub/ BQP/qpoly. To do so, we use the polynomial method to give the first correct proof of a direct product theorem for quantum search. This theorem has many other applications; for example, it can be used to fix and even improve on a flawed result of Klauck about quantum time-space tradeoffs for sorting. Third, we introduce a trace distance method for proving lower bounds on quantum one-way communication complexity. Using this method, we obtain optimal quantum lower bounds for two problems of Ambainis, for which no nontrivial lower bounds were previously known even for classical randomized protocols.

IEEE Conference on Computational Complexity (CCC)Nov 12, 200430 citations

Limitations of quantum advice and one-way communication

Scott Aaronson

Abstract: Although a quantum state requires exponentially many classical bits to de-scribe, the laws of quantum mechanics impose severe restrictions on how that state can be accessed. This paper shows in three settings that quantum messages have only limited advantages over classical ones. First, we show that BQP/qpoly⊆PP/poly, where BQP/qpoly is the class of problems solvable in quantum polynomial time, given a polynomial-size “quantum advice state ” that depends only on the input length. This resolves a question of Buhrman, and means that we should not hope for an unrelativized separation between quantum and classical advice. Un-derlying our complexity result is a general new relation between deterministic and quantum one-way communication complexities, which applies to partial as well as total functions. Second, we construct an oracle relative to which NP 6 ⊂ BQP/qpoly. To do so, we use the polynomial method to give the first correct proof of a direct product theorem for quantum search. This theorem has other applications; for example, it can be used to fix a result of Klauck about quantum time-space tradeoffs for sorting. Third, we introduce a new trace distance method for proving lower bounds on quan-tum one-way communication complexity. Using this method, we obtain optimal quantumAbstract: Although a quantum state requires exponentially many classical bits to de-scribe, the laws of quantum mechanics impose severe restrictions on how that state can be accessed. This paper shows in three settings that quantum messages have only limited advantages over classical ones. First, we show that BQP/qpoly⊆PP/poly, where BQP/qpoly is the class of problems solvable in quantum polynomial time, given a polynomial-size “quantum advice state ” that depends only on the input length. This resolves a question of Buhrman, and means that we should not hope for an unrelativized separation between quantum and classical advice. Un-derlying our complexity result is a general new relation between deterministic and quantum one-way communication complexities, which applies to partial as well as total functions. Second, we construct an oracle relative to which NP 6 ⊂ BQP/qpoly. To do so, we use the polynomial method to give the first correct proof of a direct product theorem for quantum search. This theorem has other applications; for example, it can be used to fix a result of Klauck about quantum time-space tradeoffs for sorting. Third, we introduce a new trace distance method for proving lower bounds on quan-tum one-way communication complexity. Using this method, we obtain optimal quantum

Conference on Computational ComplexityJun 21, 200413 citations

A Qubit, a Coin, and an Advice String Walk Into a Relational Problem

Scott Aaronson, Harry Buhrman, William Kretschmer

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arXiv:2302.10332Leibniz international proceedings in informaticsFeb 20, 20236 citations

One-Way Communication Complexity and the Nečiporuk Lower Bound on Formula Size

Hartmut Klauck

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SIAM Journal on ComputingJan 1, 200731 citations

Two Results about Quantum Messages

Hartmut Klauck, Supartha Podder

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arXiv:1402.4312Feb 18, 20140 citations

Classical vs. quantum communication in XOR games

Marius Junge, Carlos Palazuelos, Ignacio Villanueva

In this work we introduce an intermediate setting between quantum nonlocality and communication complexity problems. More precisely, we study the value of XOR games $G$ when Alice and Bob are allowed to use a limited amount of one-way classical communication $ω_{o.w.-c}(G)$ (resp. one-way quantum communication $ω_{o.w.-c}^*(G)$ ), where $c$ denotes the number of bits (resp. qubits). The key quantity here is the quotient $ω_{o.w.-c}^*(G)/ω_{o.w.-c}(G)$ . We provide a universal way to obtain Bell inequality violations of general Bell functionals from XOR games for which the quotient $ω_{o.w.-c}^*(G)/ω_{o.w.-2c}(G)$ is larger than 1. This allows, in particular, to find (unbounded) Bell inequality violations from communication complexity problems in the same spirit as the recent work by Buhrman et al. (2016). We also provide an example of a XOR game for which the previous quotient is optimal (up to a logarithmic factor) in terms of the amount of information $c$ . Interestingly, this game has only polynomially many inputs per player. For the related problem of separating the classical vs quantum communication complexity of a function, the known examples attaining exponential separation require exponentially many inputs per party.In this work we introduce an intermediate setting between quantum nonlocality and communication complexity problems. More precisely, we study the value of XOR games $G$ when Alice and Bob are allowed to use a limited amount of one-way classical communication $ω_{o.w.-c}(G)$ (resp. one-way quantum communication $ω_{o.w.-c}^*(G)$), where $c$ denotes the number of bits (resp. qubits). The key quantity here is the quotient $ω_{o.w.-c}^*(G)/ω_{o.w.-c}(G)$. We provide a universal way to obtain Bell inequality violations of general Bell functionals from XOR games for which the quotient $ω_{o.w.-c}^*(G)/ω_{o.w.-2c}(G)$ is larger than 1. This allows, in particular, to find (unbounded) Bell inequality violations from communication complexity problems in the same spirit as the recent work by Buhrman et al. (2016). We also provide an example of a XOR game for which the previous quotient is optimal (up to a logarithmic factor) in terms of the amount of information $c$. Interestingly, this game has only polynomially many inputs per player. For the related problem of separating the classical vs quantum communication complexity of a function, the known examples attaining exponential separation require exponentially many inputs per party.

arXiv:1706.02653Jun 8, 20170 citations

\mathsf{BQP/qpoly}\subseteq\mathsf{PP/poly}

, where

\mathsf{BQP/qpoly}

\mathsf{NP}\not \subset \mathsf{BQP/qpoly}

f:\{0,1\}^n\times\{0,1\}^m\to\{0,1\}

we have

D^{A\to B}(f)\leq O(Q^{A\to B,*}(f)\cdot m)

. This bound was previously known for total functions, while for partial functions this improves on results by Aaronson, in which either a log-factor on the right hand is present, or the left hand side is

R^{A\to B}(f)

, and in which also no entanglement is allowed. In our second contribution we investigate the power of quantum proofs over classical proofs. We give the first example of a scenario, where quantum proofs lead to exponential savings in computing a Boolean function. The previously only known separation between the power of quantum and classical proofs is in a setting where the input is also quantum. We exhibit a partial Boolean function

f

, such that there is a one-way quantum communication protocol receiving a quantum proof (i.e., a protocol of type QMA) that has cost

O(\log n)

for

f

, whereas every one-way quantum protocol for

f

receiving a classical proof (protocol of type QCMA) requires communication

Ω(\sqrt n/\log n)

In this work we introduce an intermediate setting between quantum nonlocality and communication complexity problems. More precisely, we study the value of XOR games

G

when Alice and Bob are allowed to use a limited amount of one-way classical communication

ω_{o.w.-c}(G)

(resp. one-way quantum communication

ω_{o.w.-c}^*(G)

), where

c

denotes the number of bits (resp. qubits). The key quantity here is the quotient

ω_{o.w.-c}^*(G)/ω_{o.w.-c}(G)

. We provide a universal way to obtain Bell inequality violations of general Bell functionals from XOR games for which the quotient

ω_{o.w.-c}^*(G)/ω_{o.w.-2c}(G)

is larger than 1. This allows, in particular, to find (unbounded) Bell inequality violations from communication complexity problems in the same spirit as the recent work by Buhrman et al. (2016). We also provide an example of a XOR game for which the previous quotient is optimal (up to a logarithmic factor) in terms of the amount of information

c

. Interestingly, this game has only polynomially many inputs per player. For the related problem of separating the classical vs quantum communication complexity of a function, the known examples attaining exponential separation require exponentially many inputs per party.