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Cited by 899Quantum Computing Algorithms and ArchitectureQuantum Information and CryptographyQuantum Mechanics and Applications

Quantum computers and dissipation

Open access
G. Massimo Palma, Kalle‐Antti Suominen, ARTUR K. EKERT·1996·Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences
DOI 10.1098/rspa.1996.0029
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Abstract

We analyse dissipation in quantum computation and its destructive impact on efficiency of quantum algorithms. Using a general model of decoherence, we study the time evolution of a quantum register of arbitrary length coupled with an environment of arbitrary coherence length. We discuss relations between decoherence and computational complexity and show that the quantum factorization algorithm must be modified in order to be regarded as efficient and realistic.

Verified 2026-10-06 · 2 references not checked yet

AI snapshot

Written from the paper’s full text.

TL;DR

The paper analyzes how dissipation/decoherence in quantum computers destroys quantum interference and entanglement, causing the probability of successful computation to decrease exponentially with input size \(L\). It argues that Shor’s factoring algorithm must be modified, e.g. supplemented with exponentially efficient quantum error correction, to be regarded as efficient and realistic.

Key findings

  • If environmental coupling makes the error rate grow as \(\epsilon(L)=1-A\exp(-\alpha L)\), then the required number of repetitions grows as \(k(L)\approx \exp(\alpha L)\), so the randomized factoring algorithm is no longer efficient.
  • In the independent-reservoir limit, the worst-case density-matrix elements decay as \(\exp\{-L\Gamma(t)\}\); for reservoirs with large coherence length, some off-diagonal elements decay as \(\exp\{-\mathrm{poly}(L)\Gamma(t)\}\), which can be more of a hindrance than a help unless selected coherences can be supported.
  • Single-qubit decoherence has three regimes: a “quiet” regime for \(t<\omega_c^{-1}\), a quantum regime for \(\omega_c^{-1}<t<T^{-1}\), and a thermal regime for \(t>T^{-1}\). In one dimension at low temperature, \(\Gamma(t)\sim \omega_c^2 t^2\), then \(\sim 2\ln \omega_c t\), then \(\sim Tt\); in three dimensions the decoherence saturates to a value determined by \(\eta=\omega_c/T\).
  • For two qubits, collective effects produce “subdecoherent” and “superdecoherent” states, with collective decay constants \(\Gamma_\pm(R,t)\); the paper distinguishes superdecoherence from superradiance and notes that some superradiant states are subdecoherent.

Methods

  • Theoretical study of an \(L\)-qubit quantum register coupled to a thermal reservoir, using a decoherence model with Hamiltonian \(H\) containing free qubit, free environment, and qubit-environment interaction terms.
  • Single-qubit decoherence is treated semiclassically via stochastic fluctuating fields and numerically simulated, then fully quantized using field modes, displacement operators, and exact expressions for the reduced density-matrix coherence.
  • The analysis is extended to two qubits and then to \(L\) qubits, comparing independent reservoirs with shared reservoirs and position-dependent couplings; complexity consequences are assessed through randomized-algorithm error amplification.

Limitations

  • The paper states that its model does not describe \(T_1\) processes: populations are unaffected and there is no energy exchange between qubit and environment; it notes that at longer timescales \(T_1\) processes would remove residual coherence in the three-dimensional case.
  • The paper states that the semiclassical approach excludes quantum features, most notably qubit-environment entanglement, and that a fully quantum approach is needed for thermal and vacuum fluctuations.
  • Evident from the text: the conclusion that any thermal-reservoir decoherence model leads to the same complexity conclusion is presented as a plausibility argument, not a proof; the analysis assumes specific coupling/density-of-states forms and does not include direct two-qubit interactions.

Model-written summary. Check it against the paper before relying on it.

Integrity signals

Registry

Crossref · 10.1098/rspa.1996.0029

Cites

13 of 15 references verified0 citation errors · 0 not checkable

Statistics

Nothing to recompute.

Claims the paper makes

0 of 14 cited sources checked · 0 supported · 0 flagged

  • Findingnone given
    § 1 Introduction · p. 2

    The error rate due to interaction with the environment, a thermal reservoir, grows as an exponential function of L regardless of the type and strength of the computer-environment coupling.

    “This is because the error rate, which is due to the interaction with the environment (a thermal reservoir), grows as an exponential function of L regardless the type and the strength of the computer - environment coupling.”
    • Also stated in: § 2 Errors in Randomized Algorithms · p. 3 · boosted
      “Indeed, our results show that the computer-environment interaction leads to this unwelcome exponential increase of the error rate.”
    Quote in the paperSubstantial claim
    • The authors state this as an assumption: The environment or reservoir is modeled as a thermal reservoir.
  • Findingcitation
    § 1 Introduction · p. 2

    In the limit of a short correlation length, i.e. coupling to independent reservoirs, the results coincide with those by Unruh (1995).

    “In the limit of a short correlation length (i.e. coupling to the independent reservoirs) our results coincide with those by Unruh (1995).”
    Quote in the paperNot a substantial claim0 of 1 sources checked
    • cites Unruh 1995

      Not checked in the free run.

  • Findingreasoning
    § 4.2 Fully Quantised Model · p. 10

    Thermal fluctuations can affect the qubit dynamics only for times longer than the characteristic thermal frequency T.

    “Let us first note that thermal fluctuations can affect the qubit dynamics only for times longer than the characteristic thermal frequency T .”
    Quote in the paperSubstantial claim
    • The authors state this as an assumption: The dynamics of a single qubit plus environment is described by Hamiltonian (12).
  • Findingreasoning
    § 4.2 Fully Quantised Model · p. 11

    In the one-dimensional field case in the low-temperature limit, Γ(t) ∼ Tt for t > T^-1.

    “Expression (23) reduces to Γ( t ) ∼ ω 2 c t 2 for t &lt; ω -1 c , Γ( t ) ∼ 2 ln ω c t for ω -1 c &lt; t &lt; T -1 , and Γ( t ) ∼ Tt for t &gt; T -1 .”
    Quote in the paperSubstantial claim
    • The authors state this as an assumption: The frequency dependence of the density of states and coupling is assumed to be (dk/dω)G(ω)|g(ω)|^2 ∝ ω^n e^{-ω/ωc}, with n depending on the number of dimensions.
  • Findingreasoning
    § 5 Decoherence of Quantum Registers · p. 16

    In the independent reservoir case, in the worst case the density matrix elements decay as exp{-L Γ(t)}.

    “In the independent reservoir case therefore in the worst case the density matrix elements decay as exp {-L Γ( t ) } .”
    Quote in the paperSubstantial claim
  • May combine two claims. 25
  • Classified as probably not a substantial claim. 7

  • May combine two claims.

    Findingreasoning
    § 4.2 Fully Quantised Model · p. 8

    Since [σ_z, H] = 0, the populations of the qubit density matrix are not affected by the environment, and the coupling with the environment in the model simply erodes quantum coherence.

    “Since [ σ z , H ] = 0 the populations of the qubit density matrix, ρ ( t ) = Tr R ̺ ( t ) are not affected by the environment and the coupling with the environment in our model simply erodes quantum coherence.”
    Quote in the paperSubstantial claim
    • The authors state this as an assumption: The dynamics of a single qubit plus environment is described by Hamiltonian (12).

  • May combine two claims.

    Findingnone given
    § 6 Discussion and Conclusions · p. 17

    States for which ia = ja and ib = jb do not get entangled with the reservoir and therefore maintain their coherence.

    “It is obvious that states for which i a = j a and i b = j b do not get entangled with the reservoir and therefore they mantain their coherence.”
    Quote in the paperSubstantial claim

  • May combine two claims.

    Findingreasoning
    § 5 Decoherence of Quantum Registers · p. 14

    Matrix elements of the form ρ_{ia ia, ib jb} show no collective decay; for ib = jb they are constant, while for ib ≠ jb they decay as e^{-Γ(t)}.

    “Matrix elements of the form ρ i a i a ,i b j b show no collective decay, in fact”
    • Also stated in: § 5 Decoherence of Quantum Registers · p. 14 · plain
      “which for i b = j b reduces to ρ i a i a ,i b i b ( t ) = ρ i a i a ,i b i b (0) while for i b = j b we have ρ i a i a ,i b j b ( t ) = ρ i a i a ,i b j b (0) e -Γ( t ) .”
    Quote in the paperSubstantial claim

  • May combine two claims.

    Findingreasoning
    § 4.2 Fully Quantised Model · p. 8

    In the fully quantised model there is no exchange of energy between qubit and environment and consequently no T1 type of decay takes place.

    “This means that also in our fully quantised model there is no exchange of energy between qubit and environment and consequently no T 1 type of decay takes place.”
    Quote in the paperSubstantial claim
    • The authors state this as an assumption: The dynamics of a single qubit plus environment is described by Hamiltonian (12).

  • Probably not a substantial claim: may not pass S1 (asserted), S3 (load-bearing), S4 (based).

    Findingnone given
    § 5 Decoherence of Quantum Registers · p. 14

    Collective decay is apparent in matrix elements of the form ρ_{ia ja, ib jb}(t) with ia ≠ ja and ib ≠ jb.

    “Collective decay is instead apparent in the matrix elements of the form ρ i a j a ,i b j b ( t ) with i a = j a , i b = j b :”
    Quote in the paperNot a substantial claim

  • Probably not a substantial claim: may not pass S3 (load-bearing).

    Findingown data
    § 4.2 Fully Quantised Model · p. 11

    For a single qubit in a one-dimensional field with ωc/T = 100, the three decoherence regimes can be identified in Fig. 4.

    “These three regimes can be easily identified in Fig.(4), which shows the decoherence of a single qubit induced by a one-dimensional field for the particular choice ω c /T = 100.”
    Quote in the paperNot a substantial claim
    • The authors state this as an assumption: The frequency dependence of the density of states and coupling is assumed to be (dk/dω)G(ω)|g(ω)|^2 ∝ ω^n e^{-ω/ωc}, with n depending on the number of dimensions.

  • May combine two claims.

    Interpretationreasoning
    § 1 Introduction · p. 2

    Technological progress in suppressing dissipation will help to factor bigger and bigger numbers but will not change the exponential increase of the error rate.

    “Any technological progress in suppressing dissipation will, of course, help to factor bigger and bigger numbers but will not change an exponential increase of the error rate.”
    Quote in the paperSubstantial claim

  • Probably not a substantial claim: may not pass S3 (load-bearing). May combine two claims.

    Interpretationnone given
    § 1 Introduction · p. 2

    The analysis covers cases where qubits interact with a single reservoir and the distance between the qubits is comparable with the correlation length of the reservoir.

    “Our analysis is quite general and covers cases where qubits interact with a single reservoir and the distance between the qubits is comparable with the correlation length of the reservoir.”
    Quote in the paperNot a substantial claim

  • Probably not a substantial claim: may not pass S3 (load-bearing).

    Premisecitation
    § 1 Introduction · p. 1

    Shor has shown that quantum computers require an execution time that grows only as a polynomial function of L, approximately L^2.

    “In contrast Shor (1994) has shown that quantum computers require an execution time that grows only as a polynomial function of L ( ≈ L 2 ).”
    Quote in the paperNot a substantial claim0 of 1 sources checked
    • cites Shor 1994

      Not checked in the free run.

  • Probably not a substantial claim: may not pass S3 (load-bearing). May combine two claims.

    Study design and conductnone given
    § 4.2 Fully Quantised Model · p. 10

    Equation (19) is exact; no approximation has been made to obtain it.

    “Equation (19) is exact , i.e. no approximation has been made to obtain it.”
    Quote in the paperNot a substantial claim
    • The authors state this as an assumption: The dynamics of a single qubit plus environment is described by Hamiltonian (12).

  • May combine two claims.

    Interpretationreasoning
    § 4.2 Fully Quantised Model · p. 8

    The absence of T1 decay is not a limitation of the model because the model describes adequately the decoherence mechanism, which usually takes place on a shorter timescale than energy dissipation.

    “This is not a limitation of the model since, as will became clear in the following, it describes adequately the decoherence mechanism which usually takes place on a shorter timescale than the energy dissipation.”
    Quote in the paperSubstantial claim

  • May combine two claims.

    Study design and conductnone given
    § 4.2 Fully Quantised Model · p. 8

    The model is exactly soluble and allows a clear analysis of the mechanism of entanglement between qubit and environment.

    “Furthermore this model is exactly soluble and allows a clear analysis of the mechanism of entanglement between qubit and environment which is believed to be at the core of most decoherence processes.”
    Quote in the paperSubstantial claim
    • The authors state this as an assumption: The dynamics of a single qubit plus environment is described by Hamiltonian (12).

  • May combine two claims.

    Premisenone given
    § 3 Decoherence · p. 4

    When a reservoir is in a thermal state, coherences change on a much shorter time scale than populations, and decoherence is responsible for destroying quantum parallelism.

    “When a reservoir is in a thermal state the coherences change on a much shorter time scale than populations and it is decoherence which is responsible for destroying quantum parallelism.”
    Quote in the paperSubstantial claim
    • The authors state this as an assumption: The environment or reservoir is modeled as a thermal reservoir.

  • May combine two claims.

    Interpretationreasoning
    § 5 Decoherence of Quantum Registers · p. 13

    In the limit ra ≈ rb, the subdecoherent state |Φ(-)> has qubits not entangled with the field, while the superdecoherent state |Φ(+)> has qubits collectively entangled with a coherent state whose amplitude is twice that of the single-qubit case.

    “In particular, in the limit r a ≈ r b we have ξ a k ∼ ξ b k and therefore in the 'subdecoherent' state | Φ ( -) 〉 the qubits are not entangled with the field, while in the 'superdecoherent' state | Φ (+) 〉 the qubits are collectively entangled with a coherent state with an amplitude which is twice that of the single qubit case.”
    Quote in the paperSubstantial claim

  • May combine two claims.

    Interpretationnone given
    § 5 Decoherence of Quantum Registers · p. 15

    Superdecoherence requires the qubit to be in a region of space smaller than the coherence length of the environment, a much more stringent condition than that needed for superradiance, namely that the atoms are at a distance smaller than the wavelength of the resonant field modes.

    “Indeed the conditions for the two processes to occur are quite different: superdecoherence requires the qubit to be in a region of space smaller than the coherence length of the environment, which is a much more stringent condition that the one needed for superradiance to take place, i.e. that the atoms are at a distance smaller than the wavelength of the resonant field modes.”
    Quote in the paperSubstantial claim

  • May combine two claims.

    Interpretationreasoning
    § 6 Discussion and Conclusions · p. 18

    The ratio t/τ, which depends on the technology employed, determines the limits of the algorithm, and it is unrealistic to assume that this ratio can be made infinite.

    “Thus the ratio t/τ , which depends on the technology employed, determines the limits of the algorithm and it is unrealistic to assume that this ratio can be made infinite.”
    Quote in the paperSubstantial claim

  • May combine two claims.

    Interpretationnone given
    § 6 Discussion and Conclusions · p. 17

    For reservoirs with a large coherent length the fastest decoherence will be proportional to exp(-poly(L)Γ'(t)), leading to the same unwelcome exponential increase of the error rate.

    “For reservoirs with a large coherent length the fastest decoherence will be proportional to exp( -poly ( L )Γ ′ ( t )) leading to the same unwelcome exponential increase of the error rate.”
    Quote in the paperNot a substantial claim

  • May combine two claims.

    Premisecitation
    § 2 Errors in Randomized Algorithms · p. 3

    Shor's quantum factoring algorithm in a single run provides factors of N with probability 1 - ǫ(L), and the probability of error ǫ(L) grows only as a polynomial function of L.

    “Indeed, Shor's quantum factoring algorithm is of this type; in a single run it provides factors of N with probability 1 -ǫ ( L ) and the probability of error ǫ ( L ) grows only as a polynomial function of L .”
    Quote in the paperSubstantial claim0 of 1 sources checked
    • cites Shor 1994

      Not checked in the free run.

  • Probably not a substantial claim: may not pass S3 (load-bearing). May combine two claims.

    Interpretationnone given
    § 5 Decoherence of Quantum Registers · p. 15

    Superdecoherence is due to collective entanglement between qubits and environment with no exchange of energy, while superradiance is a process of collective radiation by a group of closely spaced atoms.

    “While superdecoherence is due to collective entanglement between qubits and environment with no exchange of energy, superradiance is a process of collective radiation by a group of closely spaced atoms.”
    Quote in the paperNot a substantial claim

  • May combine two claims.

    Interpretationown data
    § 6 Discussion and Conclusions · p. 16

    Decoherence destroys quantum interference and entanglement in quantum computers, thus decreasing the probability of successful computation exponentially with the input size L.

    “Our analysis shows that decoherence destroys quantum interference and entanglement in quantum computers, thus decreasing the probability of successful computation exponentially with the input size L .”
    Quote in the paperSubstantial claim

  • Probably not a substantial claim: may not pass S3 (load-bearing), S4 (based).

    Interpretationnone given
    § 5 Decoherence of Quantum Registers · p. 15

    Some superradiant states are subdecoherent.

    “Furthermore, it must be noted that some superradiant states are subdecoherent.”
    Quote in the paperNot a substantial claim

  • May combine two claims.

    Interpretationreasoning
    § 6 Discussion and Conclusions · p. 16

    In the independent reservoirs limit the fastest decoherence is proportional to exp(-L Γ(t)), whereas for reservoirs with a large coherence length there are off-diagonal elements which decay as exp(-poly(L)Γ(t)).

    “In the independent reservoirs limit the fastest decoherence is proportional to exp( -L Γ( t )) whereas for the reservoirs with a large coherence length there are off-diagonal elements which decay as exp( -poly ( L )Γ( t )).”
    Quote in the paperSubstantial claim

  • May combine two claims.

    Premisenone given
    § 3 Decoherence · p. 4

    When a quantum computer is in contact with an environment modeled as a thermal reservoir, the resulting dissipation destroys coherences and changes populations.

    “When a quantum computer is in contact with an environment, here modeled as a thermal reservoir, the resulting dissipation destroys the coherences and changes the populations (the diagonal elements).”
    Quote in the paperSubstantial claim
    • The authors state this as an assumption: The environment or reservoir is modeled as a thermal reservoir.

  • May combine two claims.

    Interpretationreasoning
    § 2 Errors in Randomized Algorithms · p. 3

    If the interaction with the environment causes an exponential increase of the error rate, ǫ(L) = 1 - A exp(-αL), then k(L) ≈ exp(αL) and the randomized algorithm cannot be regarded as efficient any more.

    “If the interaction with the environment causes an exponential increase of the error rate i.e. ǫ ( L ) = 1 -A exp( -αL ), where A and α are positive constants, then k ( L ) ≈ exp( αL ) and the randomized algorithm cannot be regarded as efficient any more!”
    Quote in the paperSubstantial claim
    • The authors state this as an assumption: In the amplification argument, the error rate ǫ is independent of the input.

  • May combine two claims.

    Premisenone given
    § 2 Errors in Randomized Algorithms · p. 3

    Errors in Shor's algorithm are of purely mathematical origin and correspond to random decisions that rely on the prime number theorem.

    “However, errors in Shor's algorithm are of purely 'mathematical' origin and correspond to the random decisions which rely on the prime number theorem (i.e. on the distribution of prime numbers).”
    Quote in the paperNot a substantial claim

  • May combine two claims.

    Interpretationown data
    § 2 Errors in Randomized Algorithms · p. 3

    Shor's algorithm, as it stands at the moment, is not robust to environmental noise and requires modifications.

    “Shor's algorithm, as it stands at the moment, is not robust to the environmental noise and requires modifications.”
    Quote in the paperSubstantial claim

  • May combine two claims.

    Interpretationcitation
    § 4.2 Fully Quantised Model · p. 12

    The difference in behaviour between one- and three-dimensional fields is reminiscent of dimensional differences in models of phase transitions and is due to the suppressed influence of low-frequency fluctuations in three dimensions.

    “This difference in behaviour is reminiscent of the dimensional differences observed in various models of phase transitions (Ma 1976) and is due to the suppressed influence of low frequency fluctuations in three dimensions.”
    Quote in the paperNot a substantial claim0 of 1 sources checked
    • cites Ma 1976

      Not checked in the free run.

    • The authors state this as an assumption: The frequency dependence of the density of states and coupling is assumed to be (dk/dω)G(ω)|g(ω)|^2 ∝ ω^n e^{-ω/ωc}, with n depending on the number of dimensions.

  • May combine two claims.

    Interpretationreasoning
    § 6 Discussion and Conclusions · p. 16

    For reservoirs with a large coherence length, some coherences are not affected by decoherence while some others are destroyed more rapidly.

    “In this case while some coherences are not affected by the decoherence some others are destroyed more rapidly.”
    Quote in the paperSubstantial claim

  • May combine two claims.

    Interpretationreasoning
    § 1 Introduction · p. 2

    Any model of dissipation will lead to errors that grow exponentially with L, as supported by a plausibility argument.

    “Also we provide a plausibility argument that any model of dissipation will lead to errors that grow exponentially with L .”
    Quote in the paperNot a substantial claim
29 of 29

  • The authors state this as an assumption.

    The environment or reservoir is modeled as a thermal reservoir.

    “When a quantum computer is in contact with an environment, here modeled as a thermal reservoir, the resulting dissipation destroys the coherences and changes the populations (the diagonal elements).”

  • The authors state this as an assumption.

    The quantised environment is modeled as a continuum of field modes.

    “The quantised environment is modeled as a continuum of field modes.”

  • The authors state this as an assumption.

    At time t = 0 the combined qubit-environment state is assumed to have a specified product form involving the thermal density matrix of each field mode.

    “The state of the combined system i.e. qubit + environment is described by a density operator ̺ ( t ) which at time t = 0 is assumed to be in the form”

  • The authors state this as an assumption.

    The dynamics of a single qubit plus environment is described by Hamiltonian (12).

    “We assume that the dynamics of a single qubit + environment is described by the following Hamiltonian”

  • The authors state this as an assumption.

    In the amplification argument, the error rate ǫ is independent of the input.

    “Assuming that ǫ is independent of the input the success probability of the algorithm can be amplified (boosted) arbitrarily close to 1 by repeating the computation several times.”

  • The authors state this as an assumption.

    The frequency dependence of the density of states and coupling is assumed to be (dk/dω)G(ω)|g(ω)|^2 ∝ ω^n e^{-ω/ωc}, with n depending on the number of dimensions.

    “Therefore we assume ( dk/dω ) G ( ω ) | g ( ω ) | 2 ∝ ω n e -ω/ω c .”

  • The authors state this as an assumption.

    All coherences are equally important.

    “Assuming that all coherences are equally important and that for the complexity analysis we choose the worst case (i.e. the off-diagonal element which decays at the fastest rate)”

  • The authors state this as an assumption.

    For the complexity analysis, the worst case is chosen, i.e. the off-diagonal element which decays at the fastest rate.

    “Assuming that all coherences are equally important and that for the complexity analysis we choose the worst case (i.e. the off-diagonal element which decays at the fastest rate)”

  • The authors state this as an assumption.

    All modes of the field are in thermal equilibrium.

    “We can formulate in more rigorous terms the analysis we have just sketched above by taking into account all the modes of the field in thermal equilibrium.”

  • The authors state this as an assumption.

    The position-dependent couplings have the form g_a k = g_k e^{i k·r_a} and g_b k = g_k e^{i k·r_b}.

    “If we suppose g a k = g k e i k · r a and g b k = g k e i k · r b , then, in the continuum limit, the collective decay rates are”

  • The authors state this as an assumption.

    There is no direct interaction between the two qubits.

    “In (25) we are not considering any direct interaction between the two qubits which would be necessary for the conditional dynamics (Barenco et al 1995).”
11 of 11

Claims other papers make about this one

Each is checked against this paper’s own text: does it say what the citing paper says it does?

  • Noise, decoherence, and manufacturing problems make constructing large-scale quantum computers difficult.

    “Noise, decoherence, and manufacturing problems make constructing large-scale quantum computers difficult 9-13 .”

    From Experimental realization of a quantum algorithm (1998) · cites this paper as [13]

Self-check

Percentages, sums and reported differences re-computed from the paper’s own numbers.

Each contradiction between two passages listed here was confirmed by an independent check.

  • Conflicting year for the Deutsch and Jozsa citation

    Minor

    The text cites Deutsch and Jozsa as a 1992 work, but the corresponding reference lists 1989.

1 of 1

Cites

Flagged first, then by citations

Research works printed but not linked: 1

  1. Lenstra A. K. Lenstra H. W. Jr Manasse M. S. & Pollard J. M. 1990 Proc. 22th ACM Symp. j on the Theory of

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Related

Authors
Palma, G. M., Suominen, K., & EKERT, A. K.
Affiliations
  • University of Oxford (GB)
  • University of Helsinki (FI)
ORCID
Venue
Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences
Publisher
The Royal Society
Published
1996-12-31
Volume
452
Issue
1946
Pages
567-584
ISSN
  • 1364-5021 (ISSN-L)
  • 1471-2946
DOI
10.1098/rspa.1996.0029
OpenAlex
W2157754228
Cited by
899
References
15
Citations by year
2026: 10 · 2025: 25 · 2024: 22 · 2023: 26 · 2022: 29 · 2021: 22 · 2020: 23 · 2019: 32 · 2018: 34 · 2017: 35 · 2016: 38 · 2015: 31 · 2014: 29 · 2013: 37 · 2012: 25
FWCI
26.76
Type
journal article
Language
en
Open access
Locations
  • Proceedings of the Royal Society A Mathematical Physical and Engineering Sciences (publishedVersion)
  • arXiv (Cornell University) (submittedVersion)
  • CERN Document Server (European Organization for Nuclear Research) (submittedVersion)
  • arXiv (Cornell University)
Topics
  • Physical Sciences › Computer Science › Artificial Intelligence › Quantum Computing Algorithms and Architecture
  • Physical Sciences › Computer Science › Artificial Intelligence › Quantum Information and Cryptography
  • Physical Sciences › Physics and Astronomy › Atomic and Molecular Physics, and Optics › Quantum Mechanics and Applications
Keywords
quantum computing, energy dissipation, decoherence, quantum registers, environment coupling, computational complexity, quantum factoring, Shor's algorithm, algorithm efficiency

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