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/9902041 · Physical Review A

Robustness of decoherence-free subspaces for quantum computation

Dave Bacon, Daniel A. Lidar, K. Birgitta Whaley

Published Sep 1, 1999• 146 citations
arXivHTMLPDF

Abstract

It was shown recently [D. A. Lidar et al., Phys. Rev. Lett. 81, 2594 (1998)] that within the framework of the semigroup Markovian master equation, decoherence-free (DF) subspaces exist which are stable to first order in time to a perturbation. Here this result is extended to the non-Markovian regime and generalized. In particular, it is shown that within both the semigroup and the non-Markovian operator sum representation, DF subspaces are stable to all orders in time to a symmetry-breaking perturbation. DF subspaces are thus ideal for quantum memory applications. For quantum computation, however, the stability result does not extend beyond the first order. Thus, to perform robust quantum computation in DF subspaces, they must be supplemented with quantum error correcting codes.

DOI
10.1103/physreva.60.1944

Related papers

Theory of decoherence-free fault-tolerant universal quantum computation

Julia Kempe, Dave Bacon, Daniel A. Lidar +1 more

Universal quantum computation on decoherence-free subspaces and subsystems (DFSs) is examined with particular emphasis on using only physically relevant interactions. A necessary and sufficient condition for the existence of decoherence-free (noiseless) subsystems in the Markovian regime is derived here for the first time. A stabilizer formalism for DFSs is then developed which allows for the explicit understanding of these in their dual role as quantum error correcting codes. Conditions for the existence of Hamiltonians whose induced evolution always preserves a DFS are derived within this stabilizer formalism. Two possible collective decoherence mechanisms arising from permutation symmetries of the system-bath coupling are examined within this framework. It is shown that in both cases universal quantum computation which always preserves the DFS (natural fault-tolerant computation) can be performed using only two-body interactions. This is in marked contrast to standard error correcting codes, where all known constructions using one- or two-body interactions must leave the code space during the on-time of the fault-tolerant gates. A further consequence of our universality construction is that a single exchange Hamiltonian can be used to perform universal quantum computation on an encoded space whose asymptotic coding efficiency is unity. The exchange Hamiltonian, which is naturally present in many quantum systems, is thus asymptotically universal.

arXiv:quant-ph/0004064•Physical Review A•Mar 20, 2001•534 citations

Decoherence-Free Subspaces for Quantum Computation

Daniel A. Lidar, Isaac L. Chuang, K. Birgitta Whaley

Decoherence in quantum computers is formulated within the semigroup approach. The error generators are identified with the generators of a Lie algebra. This allows for a comprehensive description which includes as a special case the frequently assumed spin-boson model. A generic condition is presented for errorless quantum computation: decoherence-free subspaces are spanned by those states which are annihilated by all the generators. It is shown that these subspaces are stable to perturbations and, moreover, that universal quantum computation is possible within them.

arXiv:quant-ph/9807004•Physical Review Letters•Sep 21, 1998•1,860 citations

Order-of-magnitude extension of qubit lifetimes with a decoherence-free subspace quantum error correction code

Shival Dasu, Ben Criger, Foltz, Cameron +9 more

Constructing an efficient and robust quantum memory is central to the challenge of engineering feasible quantum computer architectures. Quantum error correction codes can solve this problem in theory, but without careful design it can introduce daunting requirements that call for machines many orders of magnitude larger than what is available today. Bringing these requirements down can often be achieved by tailoring the codes to mitigate the specific forms of noise known to be present. Using a Quantinuum H1 quantum computer, we report on a robust quantum memory design using a concatenated code, with the low-level code designed to mitigate the dominant source of memory error, and a higher-level error correction scheme to enable robust computation. The resulting encoding scheme, known as a decoherence-free subspace quantum error correction code, is characterized for long probe times, and shown to extend the memory time by over an order of magnitude compared to physical qubits.

arXiv:2503.22107•Mar 28, 2025•2 citations

Quantum reservoir computing based on decoherence-free subspaces

Mikhail Altaisky, Natalia Kaputkina

The idea of coding the information in decoherence-free subspaces (DFS) to avoid decoherence in quantum computations has been receiving a lot of attention in circuit-based quantum computations, since first proposed by Zanardi & Rasetti [1] (Phys. Rev. Lett., 1997). Following [2] (Altaisky & Kaputkina, JISST, 2025), we study the possibility to harness DFS to quantum machine learning, where no explicit algorithm is present, but the quantum system learns to classify data. We have shown that a simple classification problem of distinguishing between 2-qubit product states and 2-qubit entangled states can be solved by 4-qubit quantum reservoir [3] (Ghosh, et al., npj Quantum Inf., 2019), the output of which we fed to decoherence-free subspaces.

Days on Diffraction•Jun 16, 2025•0 citations

Criteria for dynamically stable decoherence-free subspaces and incoherently generated coherences

Raisa I. Karasik, Karl-Peter Marzlin, Barry C. Sanders +1 more

We present a detailed analysis of decoherence free subspaces and develop a rigorous theory that provides necessary and sufficient conditions for dynamically stable decoherence free subspaces. This allows us to identify a special class of decoherence free states which rely on incoherent generation of coherences. We provide examples of physical systems that support such states. Our approach employs Markovian master equations and applies primarily to finite-dimensional quantum systems.

arXiv:quant-ph/0702243•Physical Review A•May 1, 2008•61 citations

Decoherence-free subspaces for multiple-qubit errors. II. Universal, fault-tolerant quantum computation

Daniel A. Lidar, Dave Bacon, Julia Kempe +1 more

Decoherence-free subspaces (DFSs) shield quantum information from errors induced by the interaction with an uncontrollable environment. Here we study a model of correlated errors forming an Abelian subgroup (stabilizer) of the Pauli group (the group of tensor products of Pauli matrices). Unlike previous studies of DFSs, this type of error does not involve any spatial symmetry assumptions on the system-environment interaction. We solve the problem of universal, fault-tolerant quantum computation on the associated class of DFSs. We do so by introducing a hybrid DFS quantum error-correcting-code approach, where errors that arise due to departure of the codewords from the DFS are corrected actively.

arXiv:quant-ph/0007013•Physical Review A•Jan 17, 2001•65 citations
It was shown recently [D. A. Lidar et al., Phys. Rev. Lett. 81, 2594 (1998)] that within the framework of the semigroup Markovian master equation, decoherence-free (DF) subspaces exist which are stable to first order in time to a perturbation. Here this result is extended to the non-Markovian regime and generalized. In particular, it is shown that within both the semigroup and the non-Markovian operator sum representation, DF subspaces are stable to all orders in time to a symmetry-breaking perturbation. DF subspaces are thus ideal for quantum memory applications. For quantum computation, however, the stability result does not extend beyond the first order. Thus, to perform robust quantum computation in DF subspaces, they must be supplemented with quantum error correcting codes.
Universal quantum computation on decoherence-free subspaces and subsystems (DFSs) is examined with particular emphasis on using only physically relevant interactions. A necessary and sufficient condition for the existence of decoherence-free (noiseless) subsystems in the Markovian regime is derived here for the first time. A stabilizer formalism for DFSs is then developed which allows for the explicit understanding of these in their dual role as quantum error correcting codes. Conditions for the existence of Hamiltonians whose induced evolution always preserves a DFS are derived within this stabilizer formalism. Two possible collective decoherence mechanisms arising from permutation symmetries of the system-bath coupling are examined within this framework. It is shown that in both cases universal quantum computation which always preserves the DFS (natural fault-tolerant computation) can be performed using only two-body interactions. This is in marked contrast to standard error correcting codes, where all known constructions using one- or two-body interactions must leave the code space during the on-time of the fault-tolerant gates. A further consequence of our universality construction is that a single exchange Hamiltonian can be used to perform universal quantum computation on an encoded space whose asymptotic coding efficiency is unity. The exchange Hamiltonian, which is naturally present in many quantum systems, is thus asymptotically universal.
Decoherence in quantum computers is formulated within the semigroup approach. The error generators are identified with the generators of a Lie algebra. This allows for a comprehensive description which includes as a special case the frequently assumed spin-boson model. A generic condition is presented for errorless quantum computation: decoherence-free subspaces are spanned by those states which are annihilated by all the generators. It is shown that these subspaces are stable to perturbations and, moreover, that universal quantum computation is possible within them.
Constructing an efficient and robust quantum memory is central to the challenge of engineering feasible quantum computer architectures. Quantum error correction codes can solve this problem in theory, but without careful design it can introduce daunting requirements that call for machines many orders of magnitude larger than what is available today. Bringing these requirements down can often be achieved by tailoring the codes to mitigate the specific forms of noise known to be present. Using a Quantinuum H1 quantum computer, we report on a robust quantum memory design using a concatenated code, with the low-level code designed to mitigate the dominant source of memory error, and a higher-level error correction scheme to enable robust computation. The resulting encoding scheme, known as a decoherence-free subspace quantum error correction code, is characterized for long probe times, and shown to extend the memory time by over an order of magnitude compared to physical qubits.
The idea of coding the information in decoherence-free subspaces (DFS) to avoid decoherence in quantum computations has been receiving a lot of attention in circuit-based quantum computations, since first proposed by Zanardi & Rasetti [1] (Phys. Rev. Lett., 1997). Following [2] (Altaisky & Kaputkina, JISST, 2025), we study the possibility to harness DFS to quantum machine learning, where no explicit algorithm is present, but the quantum system learns to classify data. We have shown that a simple classification problem of distinguishing between 2-qubit product states and 2-qubit entangled states can be solved by 4-qubit quantum reservoir [3] (Ghosh, et al., npj Quantum Inf., 2019), the output of which we fed to decoherence-free subspaces.
Decoherence-free subspaces (DFSs) shield quantum information from errors induced by the interaction with an uncontrollable environment. Here we study a model of correlated errors forming an Abelian subgroup (stabilizer) of the Pauli group (the group of tensor products of Pauli matrices). Unlike previous studies of DFSs, this type of error does not involve any spatial symmetry assumptions on the system-environment interaction. We solve the problem of universal, fault-tolerant quantum computation on the associated class of DFSs. We do so by introducing a hybrid DFS quantum error-correcting-code approach, where errors that arise due to departure of the codewords from the DFS are corrected actively.