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:quant-ph/0301063 · Physical Review Letters

Efficient Classical Simulation of Slightly Entangled Quantum Computations

Guifré Vidal

Published Oct 1, 2003• 2,368 citations
arXivHTMLPDF

Abstract

We present a classical protocol to efficiently simulate any pure-state quantum computation that involves only a restricted amount of entanglement. More generally, we show how to classically simulate pure-state quantum computations on n qubits by using computational resources that grow linearly in n and exponentially in the amount of entanglement in the quantum computer. Our results imply that a necessary condition for an exponential computational speedup (with respect to classical computations) is that the amount of entanglement increases with the size n of the computation, and provide an explicit lower bound on the required growth.

DOI
10.1103/physrevlett.91.147902

Related papers

Classical simulation of slightly entangled quantum systems and quantum computations

Guifré Vidal

We present a scheme to efficiently simulate, with a classical computer, the dynamics of multipartite quantum systems on which the amount of entanglement (or of correlations in the case of mixed-state dynamics) is conveniently restricted. This result implies that, in order for a quantum computation to achieve a significant speed-up with respect to classical computations, the amount of entanglement (correlations) in a quantum computer cannot be upper bounded, but must scale with the size of the computation. On a practical side, the proposed scheme may be used to simulate the dynamics of certain systems of interest in quantum physics, such as one-dimensional spin models.

•Jan 15, 2003•0 citations

On the role of entanglement in quantum-computational speed-up

Richard Jozsa, Noah Linden

For any quantum algorithm operating on pure states, we prove that the presence of multi‐partite entanglement, with a number of parties that increases unboundedly with input size, is necessary if the quantum algorithm is to offer an exponential speed‐up over classical computation. Furthermore, we prove that the algorithm can be efficiently simulated classically to within a prescribed tolerance η even if a suitably small amount of global entanglement is present. We explicitly identify the occurrence of increasing multi‐partite entanglement in Shor's algorithm. Our results do not apply to quantum algorithms operating on mixed states in general and we discuss the suggestion that an exponential computational speed‐up might be possible with mixed states in the total absence of entanglement. Finally, despite the essential role of entanglement for pure‐state algorithms, we argue that it is nevertheless misleading to view entanglement as a key resource for quantum‐computational power.

arXiv:quant-ph/0201143•Proceedings of the Royal Society A Mathematical Physical and Engineering Sciences•Aug 8, 2003•839 citations

The one clean qubit model without entanglement is classically simulable

Mithuna Yoganathan, Chris Cade

Entanglement has been shown to be necessary for pure state quantum computation to have an advantage over classical computation. However, it remains open whether entanglement is necessary for quantum computers that use mixed states to also have an advantage. The one clean qubit model is a form of quantum computer in which the input is the maximally mixed state plus one pure qubit. Previous work has shown that there is a limited amount of entanglement present in these computations, despite the fact that they can efficiently solve some problems that are seemingly hard to solve classically. This casts doubt on the notion that entanglement is necessary for quantum speedups. In this work we show that entanglement is indeed crucial for efficient computation in this model, because without it the one clean qubit model is efficiently classically simulable.

arXiv:1907.08224•Jul 18, 2019•4 citations

Approximate simulation of entanglement with a linear cost of communication

A. Montina

Bell's theorem implies that the outcomes of local measurements on two maximally entangled systems cannot be simulated without classical communication between the parties. The communication cost is finite for $n$ Bell states, but it grows exponentially in $n$. Three simple protocols are presented that provide approximate simulations for low-dimensional entangled systems and require a linearly growing amount of communication. We have tested them by performing some simulations for a family of measurements. The maximal error is less than $1%$ in three dimensions and grows sublinearly with the number of entangled bits in the range numerically tested. One protocol is the multidimensional generalization of the exact Toner-Bacon [Phys. Rev. Lett. 91, 187904 (2003)] model for a single Bell state. The other two protocols are generalizations of an alternative exact model, which we derive from the Kochen-Specker [J. Math. Mech. 17, 59 (1967)] scheme for simulating single-qubit measurements. These protocols can give some indication for finding optimal one-way communication protocols that classically simulate entanglement and quantum channels. Furthermore they can be useful for deciding if a quantum communication protocol provides an advantage on classical protocols.

arXiv:1107.4647•Physical Review A•Oct 5, 2011•8 citations

Role of entanglement and correlations in mixed-state quantum computation

Animesh Datta, Guifré Vidal

In a quantum computation with pure states, the generation of large amounts of entanglement is known to be necessary for a speedup with respect to classical computations. However, examples of quantum computations with mixed states are known, such as the deterministic computation with one quantum qubit (DQC1) model [Knill and Laflamme, Phys. Rev. Lett. 81, 5672 (1998)], in which entanglement is at most marginally present, and yet a computational speedup is believed to occur. Correlations, and not entanglement, have been identified as a necessary ingredient for mixed-state quantum computation speedups. Here we show that correlations, as measured through the operator Schmidt rank, are indeed present in large amounts in the DQC1 circuit. This provides evidence for the preclusion of efficient classical simulation of DQC1 by means of a whole class of classical simulation algorithms, thereby reinforcing the conjecture that DQC1 leads to a genuine quantum computational speedup.

arXiv:quant-ph/0611157•Physical Review A•Apr 11, 2007•259 citations

Simplified instantaneous non-local quantum computation with applications to position-based cryptography

Salman Beigi, Robert König

Instantaneous measurements of non-local observables between space-like separated regions can be performed without violating causality. This feat relies on the use of entanglement. Here we propose novel protocols for this task and the related problem of multipartite quantum computation with local operations and a single round of classical communication. Compared to previously known techniques, our protocols reduce the entanglement consumption by an exponential amount. We also prove a linear lower bound on the amount of entanglement required for the implementation of a certain non-local measurement. These results relate to position-based cryptography: an amount of entanglement scaling exponentially with the number of communicated qubits is sufficient to render any such scheme insecure. Furthermore, we show that certain schemes are secure under the assumption that the adversary has less entanglement than a given bound and is restricted to classical communication.

arXiv:1101.1065•New Journal of Physics•Sep 26, 2011•121 citations
We present a classical protocol to efficiently simulate any pure-state quantum computation that involves only a restricted amount of entanglement. More generally, we show how to classically simulate pure-state quantum computations on n qubits by using computational resources that grow linearly in n and exponentially in the amount of entanglement in the quantum computer. Our results imply that a necessary condition for an exponential computational speedup (with respect to classical computations) is that the amount of entanglement increases with the size n of the computation, and provide an explicit lower bound on the required growth.
We present a scheme to efficiently simulate, with a classical computer, the dynamics of multipartite quantum systems on which the amount of entanglement (or of correlations in the case of mixed-state dynamics) is conveniently restricted. This result implies that, in order for a quantum computation to achieve a significant speed-up with respect to classical computations, the amount of entanglement (correlations) in a quantum computer cannot be upper bounded, but must scale with the size of the computation. On a practical side, the proposed scheme may be used to simulate the dynamics of certain systems of interest in quantum physics, such as one-dimensional spin models.
For any quantum algorithm operating on pure states, we prove that the presence of multi‐partite entanglement, with a number of parties that increases unboundedly with input size, is necessary if the quantum algorithm is to offer an exponential speed‐up over classical computation. Furthermore, we prove that the algorithm can be efficiently simulated classically to within a prescribed tolerance η even if a suitably small amount of global entanglement is present. We explicitly identify the occurrence of increasing multi‐partite entanglement in Shor's algorithm. Our results do not apply to quantum algorithms operating on mixed states in general and we discuss the suggestion that an exponential computational speed‐up might be possible with mixed states in the total absence of entanglement. Finally, despite the essential role of entanglement for pure‐state algorithms, we argue that it is nevertheless misleading to view entanglement as a key resource for quantum‐computational power.
Entanglement has been shown to be necessary for pure state quantum computation to have an advantage over classical computation. However, it remains open whether entanglement is necessary for quantum computers that use mixed states to also have an advantage. The one clean qubit model is a form of quantum computer in which the input is the maximally mixed state plus one pure qubit. Previous work has shown that there is a limited amount of entanglement present in these computations, despite the fact that they can efficiently solve some problems that are seemingly hard to solve classically. This casts doubt on the notion that entanglement is necessary for quantum speedups. In this work we show that entanglement is indeed crucial for efficient computation in this model, because without it the one clean qubit model is efficiently classically simulable.
Bell's theorem implies that the outcomes of local measurements on two maximally entangled systems cannot be simulated without classical communication between the parties. The communication cost is finite for nnn Bell states, but it grows exponentially in nnn. Three simple protocols are presented that provide approximate simulations for low-dimensional entangled systems and require a linearly growing amount of communication. We have tested them by performing some simulations for a family of measurements. The maximal error is less than 11%1 in three dimensions and grows sublinearly with the number of entangled bits in the range numerically tested. One protocol is the multidimensional generalization of the exact Toner-Bacon [Phys. Rev. Lett. 91, 187904 (2003)] model for a single Bell state. The other two protocols are generalizations of an alternative exact model, which we derive from the Kochen-Specker [J. Math. Mech. 17, 59 (1967)] scheme for simulating single-qubit measurements. These protocols can give some indication for finding optimal one-way communication protocols that classically simulate entanglement and quantum channels. Furthermore they can be useful for deciding if a quantum communication protocol provides an advantage on classical protocols.
In a quantum computation with pure states, the generation of large amounts of entanglement is known to be necessary for a speedup with respect to classical computations. However, examples of quantum computations with mixed states are known, such as the deterministic computation with one quantum qubit (DQC1) model [Knill and Laflamme, Phys. Rev. Lett. 81, 5672 (1998)], in which entanglement is at most marginally present, and yet a computational speedup is believed to occur. Correlations, and not entanglement, have been identified as a necessary ingredient for mixed-state quantum computation speedups. Here we show that correlations, as measured through the operator Schmidt rank, are indeed present in large amounts in the DQC1 circuit. This provides evidence for the preclusion of efficient classical simulation of DQC1 by means of a whole class of classical simulation algorithms, thereby reinforcing the conjecture that DQC1 leads to a genuine quantum computational speedup.
Instantaneous measurements of non-local observables between space-like separated regions can be performed without violating causality. This feat relies on the use of entanglement. Here we propose novel protocols for this task and the related problem of multipartite quantum computation with local operations and a single round of classical communication. Compared to previously known techniques, our protocols reduce the entanglement consumption by an exponential amount. We also prove a linear lower bound on the amount of entanglement required for the implementation of a certain non-local measurement. These results relate to position-based cryptography: an amount of entanglement scaling exponentially with the number of communicated qubits is sufficient to render any such scheme insecure. Furthermore, we show that certain schemes are secure under the assumption that the adversary has less entanglement than a given bound and is restricted to classical communication.