Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:0909.3930

Computational Distinguishability of Quantum Channels

Bill Rosgen

Published Sep 22, 2009• 9 citations
arXivHTMLPDF

Abstract

The computational problem of distinguishing two quantum channels is central to quantum computing. It is a generalization of the well-known satisfiability problem from classical to quantum computation. This problem is shown to be surprisingly hard: it is complete for the class QIP of problems that have quantum interactive proof systems, which implies that it is hard for the class PSPACE of problems solvable by a classical computation in polynomial space. Several restrictions of distinguishability are also shown to be hard. It is no easier when restricted to quantum computations of logarithmic depth, to mixed-unitary channels, to degradable channels, or to antidegradable channels. These hardness results are demonstrated by finding reductions between these classes of quantum channels. These techniques have applications outside the distinguishability problem, as the construction for mixed-unitary channels is used to prove that the additivity problem for the classical capacity of quantum channels can be equivalently restricted to the mixed unitary channels.

DOI
10.48550/arxiv.0909.3930

Related papers

On the Hardness of Distinguishing Mixed-State Quantum Computations

Bill Rosgen, John Watrous

This paper considers the following problem. Two mixed-state quantum circuits Q/sub 0/ and Q/sub 1/ are given, and the goal is to determine which of two possibilities holds: (i) Q/sub 0/ and Q/sub 1/ act nearly identically on all possible quantum state inputs, or (ii) there exists some input state /spl rho/ that Q/sub 0/ and Q/sub 1/ transform into almost perfectly distinguishable outputs. This may be viewed as an abstraction of the problem that asks, given two discrete quantum mechanical processes described by sequences of local interactions, are the processes effectively the same or are they different? We prove that this promise problem is complete for the class QIP of problems having quantum interactive proof systems, and is therefore PSPACE-hard. This is in contrast to the fact that the analogous problem for classical (probabilistic) circuits is in AM, and for unitary quantum circuits is in QMA.

arXiv:cs/0407056•IEEE Conference on Computational Complexity (CCC)•Jun 28, 2005•66 citations

Additivity and distinguishability of random unitary channels

Bill Rosgen

A random unitary channel is one that is given by a convex combination of unitary channels. It is shown that the conjectures on the additivity of the minimum output entropy and the multiplicativity of the maximum output p-norm can be equivalently restated in terms of random unitary channels. This is done by constructing a random unitary approximation to a general quantum channel. This approximation can be constructed efficiently, and so it is also applied to the computational problem of distinguishing quantum circuits. It is shown that the problem of distinguishing random unitary circuits is as hard as the problem of distinguishing general mixed-state circuits, which is complete for the class of problems having quantum interactive proof systems.

arXiv:0804.1936•Journal of Mathematical Physics•Oct 1, 2008•19 citations

Additivity and distinguishability of random unitary channels

Bill Rosgen

A random unitary channel is one that is given by a convex combination of unitary channels. It is shown that the conjectures on the additivity of the minimum output entropy and the multiplicativity of the maximum output p-norm can be equivalently restated in terms of random unitary channels. This is done by constructing a random unitary approximation to a general quantum channel. This approximation can be constructed efficiently, and so it is also applied to the computational problem of distinguishing quantum circuits. It is shown that the problem of distinguishing random unitary circuits is as hard as the QIP-complete problem of distinguishing general mixed state circuits. 1

•Jan 1, 2008•15 citations

Computational distinguishability of degradable and antidegradable channels

Bill Rosgen

A channel is degradable if there exists a second channel that maps the output state of the channel to the environment state. These channels satisfy the property that the output state contains more information about the input than the environment does. A complementary class of channels is the antidegradable channels, which admit channels that map the environment state to the output state of the channel. In this paper we show that the computational problem of distinguishing two channels remains -complete when restricted to these classes of channels. This is shown using a construction of Cubitt, Ruskai, and Smith that embeds any channel into a degradable channel, and a related construction for the case of antidegradable channels.

arXiv:0911.2109•Quantum Information and Computation•Sep 1, 2010•3 citations

Distinguishing Short Quantum Computations

Bill Rosgen

Distinguishing logarithmic depth quantum circuits on mixed states is shown to be complete for $QIP$, the class of problems having quantum interactive proof systems. Circuits in this model can represent arbitrary quantum processes, and thus this result has implications for the verification of implementations of quantum algorithms. The distinguishability problem is also complete for $QIP$ on constant depth circuits containing the unbounded fan-out gate. These results are shown by reducing a $QIP$-complete problem to a logarithmic depth version of itself using a parallelization technique.

arXiv:0712.2595•Leibniz international proceedings in informatics•Dec 16, 2007•6 citations

On the undecidability of quantum channel capacities

Archishna Bhattacharyya, Arthur Mehta, Yuming Zhao

An important distinction in our understanding of capacities of classical versus quantum channels is marked by the following question: is there an algorithm which can compute (or even efficiently compute) the capacity? While there is overwhelming evidence suggesting that quantum channel capacities may be uncomputable, a formal proof of any such statement is elusive. We initiate the study of the hardness of computing quantum channel capacities. We show that, for a general quantum channel, it is QMA-hard to compute its quantum capacity, and that the entanglement-assisted zero-error capacity under some restrictions is uncomputable; indicative of the fact that quantum channel capacities may generally be undecidable.

quant-pharXiv:2601.22471•Feb 2, 2026•0 citations
The computational problem of distinguishing two quantum channels is central to quantum computing. It is a generalization of the well-known satisfiability problem from classical to quantum computation. This problem is shown to be surprisingly hard: it is complete for the class QIP of problems that have quantum interactive proof systems, which implies that it is hard for the class PSPACE of problems solvable by a classical computation in polynomial space. Several restrictions of distinguishability are also shown to be hard. It is no easier when restricted to quantum computations of logarithmic depth, to mixed-unitary channels, to degradable channels, or to antidegradable channels. These hardness results are demonstrated by finding reductions between these classes of quantum channels. These techniques have applications outside the distinguishability problem, as the construction for mixed-unitary channels is used to prove that the additivity problem for the classical capacity of quantum channels can be equivalently restricted to the mixed unitary channels.
This paper considers the following problem. Two mixed-state quantum circuits Q/sub 0/ and Q/sub 1/ are given, and the goal is to determine which of two possibilities holds: (i) Q/sub 0/ and Q/sub 1/ act nearly identically on all possible quantum state inputs, or (ii) there exists some input state /spl rho/ that Q/sub 0/ and Q/sub 1/ transform into almost perfectly distinguishable outputs. This may be viewed as an abstraction of the problem that asks, given two discrete quantum mechanical processes described by sequences of local interactions, are the processes effectively the same or are they different? We prove that this promise problem is complete for the class QIP of problems having quantum interactive proof systems, and is therefore PSPACE-hard. This is in contrast to the fact that the analogous problem for classical (probabilistic) circuits is in AM, and for unitary quantum circuits is in QMA.
A random unitary channel is one that is given by a convex combination of unitary channels. It is shown that the conjectures on the additivity of the minimum output entropy and the multiplicativity of the maximum output p-norm can be equivalently restated in terms of random unitary channels. This is done by constructing a random unitary approximation to a general quantum channel. This approximation can be constructed efficiently, and so it is also applied to the computational problem of distinguishing quantum circuits. It is shown that the problem of distinguishing random unitary circuits is as hard as the problem of distinguishing general mixed-state circuits, which is complete for the class of problems having quantum interactive proof systems.
A random unitary channel is one that is given by a convex combination of unitary channels. It is shown that the conjectures on the additivity of the minimum output entropy and the multiplicativity of the maximum output p-norm can be equivalently restated in terms of random unitary channels. This is done by constructing a random unitary approximation to a general quantum channel. This approximation can be constructed efficiently, and so it is also applied to the computational problem of distinguishing quantum circuits. It is shown that the problem of distinguishing random unitary circuits is as hard as the QIP-complete problem of distinguishing general mixed state circuits. 1
A channel is degradable if there exists a second channel that maps the output state of the channel to the environment state. These channels satisfy the property that the output state contains more information about the input than the environment does. A complementary class of channels is the antidegradable channels, which admit channels that map the environment state to the output state of the channel. In this paper we show that the computational problem of distinguishing two channels remains -complete when restricted to these classes of channels. This is shown using a construction of Cubitt, Ruskai, and Smith that embeds any channel into a degradable channel, and a related construction for the case of antidegradable channels.
Distinguishing logarithmic depth quantum circuits on mixed states is shown to be complete for QIPQIPQIP, the class of problems having quantum interactive proof systems. Circuits in this model can represent arbitrary quantum processes, and thus this result has implications for the verification of implementations of quantum algorithms. The distinguishability problem is also complete for QIPQIPQIP on constant depth circuits containing the unbounded fan-out gate. These results are shown by reducing a QIPQIPQIP-complete problem to a logarithmic depth version of itself using a parallelization technique.
An important distinction in our understanding of capacities of classical versus quantum channels is marked by the following question: is there an algorithm which can compute (or even efficiently compute) the capacity? While there is overwhelming evidence suggesting that quantum channel capacities may be uncomputable, a formal proof of any such statement is elusive. We initiate the study of the hardness of computing quantum channel capacities. We show that, for a general quantum channel, it is QMA-hard to compute its quantum capacity, and that the entanglement-assisted zero-error capacity under some restrictions is uncomputable; indicative of the fact that quantum channel capacities may generally be undecidable.