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Quantum computers and dissipation
Open accessAbstract
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.
Checked 2026-10-06 · 2 references not checked yet
Overview
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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.
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The paper states this without a source or data behind it here, so there is nothing to check it against. The quote shows it is in the paper.
In the paper
“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.”
Tests✓ Asserted: passes✓ Checkable: passes✓ Load-bearing: passes✓ Based: passes- The authors state this as an assumption: The environment or reservoir is modeled as a thermal reservoir.
- Also stated in: § 2 Errors in Randomized Algorithms · p. 3 · boosted
In the paper
“In the limit of a short correlation length (i.e. coupling to the independent reservoirs) our results coincide with those by Unruh (1995).”
What the sources say
- Not checked yetcites Unruh 1995
Strength:plain· not yet calibratedTests✓ Asserted: passes✓ Checkable: passes? Load-bearing: unclear✓ Based: passesThe paper states this without a source or data behind it here, so there is nothing to check it against. The quote shows it is in the paper.
In the paper
“Let us first note that thermal fluctuations can affect the qubit dynamics only for times longer than the characteristic thermal frequency T .”
Load-bearingStrength:plain· not yet calibratedTests✓ Asserted: passes✓ Checkable: passes✓ Load-bearing: passes✓ Based: passes- The authors state this as an assumption: The dynamics of a single qubit plus environment is described by Hamiltonian (12).
The paper states this without a source or data behind it here, so there is nothing to check it against. The quote shows it is in the paper.
In the paper
“Expression (23) reduces to Γ( t ) ∼ ω 2 c t 2 for t < ω -1 c , Γ( t ) ∼ 2 ln ω c t for ω -1 c < t < T -1 , and Γ( t ) ∼ Tt for t > T -1 .”
Tests✓ Asserted: passes✓ Checkable: passes✓ Load-bearing: passes✓ Based: passes- 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.
The paper states this without a source or data behind it here, so there is nothing to check it against. The quote shows it is in the paper.
In the paper
“In the independent reservoir case therefore in the worst case the density matrix elements decay as exp {-L Γ( t ) } .”
Tests✓ Asserted: passes✓ Checkable: passes✓ Load-bearing: passes✓ Based: passesThe paper states this without a source or data behind it here, so there is nothing to check it against. The quote shows it is in the paper.
In the paper
“As η decreases, the decay onset moves towards larger t .”
Strength:plain· not yet calibratedCausal:direct· not yet calibratedTests✓ Asserted: passes✓ Checkable: passes? Load-bearing: unclear✓ Based: passes- 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.
The paper states this without a source or data behind it here, so there is nothing to check it against. The quote shows it is in the paper.
In the paper
“Indeed the off-diagonal elements of the reduced density matrix of the qubit decay due to the fact that the overlap between the different field states with which the qubit becomes entangled diminishes in time.”
Tests✓ Asserted: passes✓ Checkable: passes✓ Load-bearing: passes✓ Based: passes- The authors state this as an assumption: The dynamics of a single qubit plus environment is described by Hamiltonian (12).
The paper states this without a source or data behind it here, so there is nothing to check it against. The quote shows it is in the paper.
In the paper
“It should be noted that in the time evolution of a single member of the ensemble the amplitude for the oscillations in s x and s y remains constant.”
Load-bearingStrength:plain· not yet calibratedTests✓ Asserted: passes✓ Checkable: passes✓ Load-bearing: passes✓ Based: passesA result the paper reports from its own data — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“However, for large η there is an intermediate region where the vacuum fluctuations dominate and the decay is roughly linear before the thermal regime takes over.”
Strength:plain· not yet calibratedCausal:direct· not yet calibratedTests✓ Asserted: passes✓ Checkable: passes? Load-bearing: unclear✓ Based: passes- 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.
The paper states this without a source or data behind it here, so there is nothing to check it against. The quote shows it is in the paper.
In the paper
“Expression (23) reduces to Γ( t ) ∼ ω 2 c t 2 for t < ω -1 c , Γ( t ) ∼ 2 ln ω c t for ω -1 c < t < T -1 , and Γ( t ) ∼ Tt for t > T -1 .”
Tests✓ Asserted: passes✓ Checkable: passes✓ Load-bearing: passes✓ Based: passes- 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.
- It may state two separate claims as one. 25
- Probably not a substantial claim. 7
The paper states this without a source or data behind it here, so there is nothing to check it against. The quote shows it is in the paper.
In the paper
“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.”
Tests✓ Asserted: passes✓ Checkable: passes✓ Load-bearing: passes✓ Based: passesIt may state two separate claims as one.
- The authors state this as an assumption: The dynamics of a single qubit plus environment is described by Hamiltonian (12).
The paper states this without a source or data behind it here, so there is nothing to check it against. The quote shows it is in the paper.
In the paper
“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.”
Tests✓ Asserted: passes✓ Checkable: passes✓ Load-bearing: passes✓ Based: passesIt may state two separate claims as one.
The paper states this without a source or data behind it here, so there is nothing to check it against. The quote shows it is in the paper.
In the paper
“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 ) .”
Load-bearingStrength:plain· not yet calibratedTests✓ Asserted: passes✓ Checkable: passes✓ Load-bearing: passes✓ Based: passesIt may state two separate claims as one.
- Also stated in: § 5 Decoherence of Quantum Registers · p. 14 · plain
The paper states this without a source or data behind it here, so there is nothing to check it against. The quote shows it is in the paper.
In the paper
“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.”
Load-bearingStrength:plain· not yet calibratedTests✓ Asserted: passes✓ Checkable: passes✓ Load-bearing: passes✓ Based: passesIt may state two separate claims as one.
- The authors state this as an assumption: The dynamics of a single qubit plus environment is described by Hamiltonian (12).
The paper states this without a source or data behind it here, so there is nothing to check it against. The quote shows it is in the paper.
In the paper
“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 :”
Strength:plain· not yet calibratedCausal:correlational· not yet calibratedTests✗ Asserted: fails✓ Checkable: passes✗ Load-bearing: fails✗ Based: failsProbably not a substantial claim. The authors may not put it forward as true. The paper’s conclusions may not depend on it. The paper may give no reason why it holds.
A result the paper reports from its own data — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“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.”
Strength:plain· not yet calibratedSingle caseTests✓ Asserted: passes✓ Checkable: passes✗ Load-bearing: fails✓ Based: passesProbably not a substantial claim. The paper’s conclusions may not depend on it.
- 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.
A conclusion the paper draws from its own results — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“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.”
Tests✓ Asserted: passes✓ Checkable: passes✓ Load-bearing: passes✓ Based: passesIt may state two separate claims as one.
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In the paper
“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.”
Strength:plain· not yet calibratedCausal:correlational· not yet calibratedTests✓ Asserted: passes✓ Checkable: passes✗ Load-bearing: fails✓ Based: passesProbably not a substantial claim. The paper’s conclusions may not depend on it. It may state two separate claims as one.
In the paper
“In contrast Shor (1994) has shown that quantum computers require an execution time that grows only as a polynomial function of L ( ≈ L 2 ).”
What the sources say
- Not checked yetcites Shor 1994
Strength:plain· not yet calibratedCausal:direct· not yet calibratedTests✓ Asserted: passes✓ Checkable: passes✗ Load-bearing: fails✓ Based: passesProbably not a substantial claim. The paper’s conclusions may not depend on it.
What the paper did — it cites no source for this, so there is none to check. The quote shows it is in the paper.
In the paper
“Equation (19) is exact , i.e. no approximation has been made to obtain it.”
Strength:plain· not yet calibratedTests✓ Asserted: passes✓ Checkable: passes✗ Load-bearing: fails✓ Based: passesProbably not a substantial claim. The paper’s conclusions may not depend on it. It may state two separate claims as one.
- The authors state this as an assumption: The dynamics of a single qubit plus environment is described by Hamiltonian (12).
A conclusion the paper draws from its own results — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“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.”
Load-bearingStrength:plain· not yet calibratedTests✓ Asserted: passes✓ Checkable: passes✓ Load-bearing: passes✓ Based: passesIt may state two separate claims as one.
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In the paper
“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.”
Load-bearingStrength:boosted· not yet calibratedTests✓ Asserted: passes✓ Checkable: passes✓ Load-bearing: passes✓ Based: passesIt may state two separate claims as one.
- The authors state this as an assumption: The dynamics of a single qubit plus environment is described by Hamiltonian (12).
The paper states this without a source or data behind it here, so there is nothing to check it against. The quote shows it is in the paper.
In the paper
“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.”
Tests✓ Asserted: passes✓ Checkable: passes✓ Load-bearing: passes✓ Based: passesIt may state two separate claims as one.
- The authors state this as an assumption: The environment or reservoir is modeled as a thermal reservoir.
A conclusion the paper draws from its own results — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“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.”
Tests✓ Asserted: passes✓ Checkable: passes✓ Load-bearing: passes✓ Based: passesIt may state two separate claims as one.
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In the paper
“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.”
Tests✓ Asserted: passes✓ Checkable: passes✓ Load-bearing: passes✓ Based: passesIt may state two separate claims as one.
A conclusion the paper draws from its own results — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“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.”
Tests✓ Asserted: passes✓ Checkable: passes✓ Load-bearing: passes✓ Based: passesIt may state two separate claims as one.
A conclusion the paper draws from its own results — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“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.”
Tests✓ Asserted: passes✓ Checkable: passes✓ Load-bearing: passes? Based: unclearIt may state two separate claims as one.
In the paper
“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 .”
What the sources say
- Not checked yetcites Shor 1994
Tests✓ Asserted: passes✓ Checkable: passes✓ Load-bearing: passes✓ Based: passesIt may state two separate claims as one.
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In the paper
“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.”
Strength:plain· not yet calibratedCausal:direct· not yet calibratedTests✓ Asserted: passes✓ Checkable: passes✗ Load-bearing: fails✓ Based: passesProbably not a substantial claim. The paper’s conclusions may not depend on it. It may state two separate claims as one.
A conclusion the paper draws from its own results — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“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 .”
Tests✓ Asserted: passes✓ Checkable: passes✓ Load-bearing: passes✓ Based: passesIt may state two separate claims as one.
A conclusion the paper draws from its own results — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“Furthermore, it must be noted that some superradiant states are subdecoherent.”
Strength:plain· not yet calibratedCausal:correlational· not yet calibratedTests✓ Asserted: passes✓ Checkable: passes✗ Load-bearing: fails✗ Based: failsProbably not a substantial claim. The paper’s conclusions may not depend on it. The paper may give no reason why it holds.
A conclusion the paper draws from its own results — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“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 )).”
Tests✓ Asserted: passes✓ Checkable: passes✓ Load-bearing: passes✓ Based: passesIt may state two separate claims as one.
The paper states this without a source or data behind it here, so there is nothing to check it against. The quote shows it is in the paper.
In the paper
“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).”
Tests✓ Asserted: passes✓ Checkable: passes✓ Load-bearing: passes✓ Based: passesIt may state two separate claims as one.
- The authors state this as an assumption: The environment or reservoir is modeled as a thermal reservoir.
A conclusion the paper draws from its own results — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“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!”
Tests✓ Asserted: passes✓ Checkable: passes✓ Load-bearing: passes✓ Based: passesIt may state two separate claims as one.
- The authors state this as an assumption: In the amplification argument, the error rate ǫ is independent of the input.
The paper states this without a source or data behind it here, so there is nothing to check it against. The quote shows it is in the paper.
In the paper
“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).”
Strength:plain· not yet calibratedCausal:direct· not yet calibratedTests✓ Asserted: passes✓ Checkable: passes? Load-bearing: unclear✓ Based: passesIt may state two separate claims as one.
A conclusion the paper draws from its own results — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“Shor's algorithm, as it stands at the moment, is not robust to the environmental noise and requires modifications.”
Tests✓ Asserted: passes✓ Checkable: passes✓ Load-bearing: passes✓ Based: passesIt may state two separate claims as one.
In the paper
“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.”
What the sources say
- Not checked yetcites Ma 1976
Strength:plain· not yet calibratedCausal:direct· not yet calibratedTests✓ Asserted: passes✓ Checkable: passes? Load-bearing: unclear✓ Based: passesIt may state two separate claims as one.
- 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.
A conclusion the paper draws from its own results — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“In this case while some coherences are not affected by the decoherence some others are destroyed more rapidly.”
Tests✓ Asserted: passes✓ Checkable: passes✓ Load-bearing: passes✓ Based: passesIt may state two separate claims as one.
A conclusion the paper draws from its own results — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“Also we provide a plausibility argument that any model of dissipation will lead to errors that grow exponentially with L .”
Tests✓ Asserted: passes✓ Checkable: passes✓ Load-bearing: passes? Based: unclearIt may state two separate claims as one.
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).”
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In the citing paper
“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]
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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
MinorThe text cites Deutsch and Jozsa as a 1992 work, but the corresponding reference lists 1989.
“(Deutsch 1985, Deutsch and Jozsa 1992, Bernstein and Vazirani 1993, Simon 1994, Shor 1994)”
“Deutsch, D. & Jozsa, R. 1989 Proc. Roy. Soc. A439 , pp. 553-558.”
Citations
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Dynamics of the dissipative two-state system
Leggett, A. J., Chakravarty, S., Dorsey, A. T., Fisher, M. P. A., Garg, A., & Zwerger, W. (1987) · Reviews of Modern Physics · doi:10.1103/revmodphys.59.1 · Cited by 5,109 · cited 1×
“Hamiltonian (12) is equivalent to the one introduced in connection with the tunnelling problem (Leggett et al. 1987, Gardiner 1991) and used to model decoherence in quantum computers (Unruh 1995).”
MethodQuantum theory, the Church–Turing principle and the universal quantum computer
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Distribution functions in physics: Fundamentals
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Rapid solution of problems by quantum computation
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Optical resonance and two-level atoms
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Optical Resonance and Two-level Atoms
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Modern Theory of Critical Phenomena
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Distribution Functions in Physics: Fundamentals
Hillery, M. S., O’Connell, R. F., Scully, M. O., & Wigner, E. P. (1997) · Springer eBooks · doi:10.1007/978-3-642-59033-7_28 · Cited by 1,803
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- 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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Dynamical analysis of Grover’s search algorithm in arbitrarily high-dimensional search spaces
Jin, W. (2015) · Quantum Information Processing · doi:10.1007/s11128-015-1164-0 · Cited by 9
Quantum Communication with Polarization-Encoded Qubits under Majorization Monotone Dynamics
Czerwiński, A. (2022) · Mathematics · doi:10.3390/math10213932 · Cited by 9
Two-party Quantum Key Agreement with Six-particle Entangled States Against Collective Noise
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Broadband noise decoherence in solid-state complex architectures
Paladino, E., D’Arrigo, A., Mastellone, A., & Falci, G. (2009) · Physica Scripta · doi:10.1088/0031-8949/2009/t137/014017 · Cited by 9
Photonic dephasing dynamics and the role of initial correlations
Raja, S. H., Athulya, K. P., Shaji, A., & Piilo, J. (2020) · Physical Review A · doi:10.1103/physreva.101.042127 · Cited by 10
Danon, J., Chatterjee, A., Gyenis, A., & Kuemmeth, F. (2021) · Applied Physics Letters · doi:10.1063/5.0073945 · Cited by 13
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Chakour, M. H. B. A., Allati, A. E., & Hassouni, Y. (2022) · Physics Letters A · doi:10.1016/j.physleta.2022.128410 · Cited by 9
Quantum control of discrete noise in Josephson qubits
D’Arrigo, A., Falci, G., Mastellone, A., & Paladino, E. (2005) · Physica E Low-dimensional Systems and Nanostructures · doi:10.1016/j.physe.2005.05.027 · Cited by 9
Functional integral treatment of some quantum nondemolition systems
Banerjee, S., & Ghosh, R. (2007) · Journal of Physics A Mathematical and Theoretical · doi:10.1088/1751-8113/40/6/006 · Cited by 9
Segnorile, H. H., & Zamar, R. C. (2011) · The Journal of Chemical Physics · doi:10.1063/1.3668559 · Cited by 9
Stavrou, V. N. (2007) · Journal of Physics Condensed Matter · doi:10.1088/0953-8984/19/18/186224 · Cited by 9
Two-qubit gates for decoherence-free qubits using a ring exchange interaction
Antonio, B., & Bose, S. (2013) · Physical Review A · doi:10.1103/physreva.88.042306 · Cited by 9
Role of the effective Hilbert-space size of the reservoir for the decoherence process
Oliveira, A. C., & de Magalhães, A. R. B. (2009) · Physical Review E · doi:10.1103/physreve.80.026204 · Cited by 9
Decoherence induced by electron accumulation in a quantum measurement of charge qubits
Lee, M., & Zhang, W.-M. (2006) · Physical Review B · doi:10.1103/physrevb.74.085325 · Cited by 9
Dynamical fidelity susceptibility of decoherence-free subspaces
Kattemölle, J., & van Wezel, J. (2019) · Physical Review A · doi:10.1103/physreva.99.062340 · Cited by 10
Charge noise induced spin dephasing in a nanowire double quantum dot with spin–orbit coupling
Li, R. (2019) · Journal of Physics Condensed Matter · doi:10.1088/1361-648x/ab4933 · Cited by 9
Mogilevtsev, D. S., Reyes‐Gómez, E., Cavalcanti, S. B., & Oliveira, L. E. (2015) · Physical Review B · doi:10.1103/physrevb.92.235446 · Cited by 9
Decoherence-Free Subspaces and Subsystems for a Collectively Depolarizing Bosonic Channel
Ball, J. L., & Banaszek, K. (2005) · Open Systems & Information Dynamics · doi:10.1007/s11080-005-5723-1 · Cited by 10
Dynamics of incompatibility of quantum measurements in open systems
Addis, C., Heinosaari, T., Kiukas, J., Laine, E.-M., & Maniscalco, S. (2016) · Physical Review A · doi:10.1103/physreva.93.022114 · Cited by 9
Mendonça, P. E. M. F., Marchiolli, M. A., & Napolitano, R. D. J. (2005) · Journal of Physics A Mathematical and General · doi:10.1088/0305-4470/38/6/l04 · Cited by 9
Enhanced quantum estimation via purification
Mack, H., Fischer, D. G., & Freyberger, M. (2000) · Physical Review A · doi:10.1103/physreva.62.042301 · Cited by 9
Dissipation-assisted quantum computation in atom-cavity systems
Beige, A., Cable, H. V., & Knight, P. L. (2003) · Proceedings of SPIE, the International Society for Optical Engineering/Proceedings of SPIE · doi:10.1117/12.497004 · Cited by 10
Vibrational coherence in electron transfer: an exactly solvable model
Creechley, J., & Dahnovsky, Y. (2003) · Chemical Physics · doi:10.1016/j.chemphys.2003.08.025 · Cited by 9
SOLENOV, D., & PRIVMAN, V. (2006) · International Journal of Modern Physics B · doi:10.1142/s0217979206034066 · Cited by 9
Using sequences of pulses to control coherence in an open quantum computing system
Qiao, B., Ruda, H. E., & Chang, J. F. (2002) · Journal of Applied Physics · doi:10.1063/1.1473678 · Cited by 9
S-matrix approach to the construction of decoherence-free subspaces
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Zeno dynamics and constraintsPreprint
Facchi, P., Marmo, G., Pascazio, S., Scardicchio, A., & Sudarshan, E. C. G. (2003) · Cited by 9
Quantum Mpemba Effect Induced by Non-Markovian Exceptional Points
Zhang, Z.-Z., Luo, H., & Wu, W. (2026) · Physical Review Letters · doi:10.1103/zjdz-rqqd · Cited by 8
Breuer, H., & Petruccione, F. (2007) · Oxford University Press eBooks · doi:10.1093/acprof:oso/9780199213900.003.04 · Cited by 8
Deterministic quantum computation with one-clean-qubit model as an open quantum system
Xuereb, J., Campbell, S., Goold, J., & Xuereb, A. (2023) · Physical Review A · doi:10.1103/physreva.107.042222 · Cited by 8
Wu, W., Luo, D.-W., & Xu, J. (2014) · Journal of Applied Physics · doi:10.1063/1.4885425 · Cited by 8
Berrada, K., & Al-Rajhi, M. A. (2017) · Quantum Information Processing · doi:10.1007/s11128-017-1686-8 · Cited by 8
Dephasing and pseudocoherent quantum dynamics in super-Ohmic environments
Nacke, P., Otterpohl, F., Thorwart, M., & Nalbach, P. (2023) · Physical Review A · doi:10.1103/physreva.107.062218 · Cited by 8
Theory of decoherence control in a fluctuating environment
Kitajima, S., Ban, M., & Shibata, F. (2010) · Journal of Physics B Atomic Molecular and Optical Physics · doi:10.1088/0953-4075/43/13/135504 · Cited by 8
Protecting two-qubit quantum states by π -phase pulses
Hu, J., Wang, X., & Kwek, L. C. (2010) · Physical Review A · doi:10.1103/physreva.82.062317 · Cited by 8
Three Different Manifestations of the Quantum Zeno Effect
Paolo Facchi, A., & Pascazio, S. (2003) · Lecture notes in physics · doi:10.1007/3-540-44874-8_7 · Cited by 8
Domínguez, F. D., González, C. E., Segnorile, H. H., & Zamar, R. C. (2016) · Physical Review A · doi:10.1103/physreva.93.022120 · Cited by 8
Preparation of a single-photon dark state in a chiral quantum system
Zhu, F., Zhao, T., Zhang, H., Li, G., & Ficek, Z. (2017) · Physical Review A · doi:10.1103/physreva.95.023817 · Cited by 8
Nonperturbative environmental influence on dephasing
Palm, T., & Nalbach, P. (2017) · Physical Review A · doi:10.1103/physreva.96.032105 · Cited by 8
Merkli, M., Berman, G. P., Borgonovi, F., & Gebresellasie, K. (2011) · Quantum Information and Computation · doi:10.26421/qic11.5-6-3 · Cited by 8
Quantitative Treatment of Decoherence
Fedichkin, L. E., & Vladimir Privman, A. (2009) · Topics in applied physics · doi:10.1007/978-3-540-79365-6_8 · Cited by 8
Probing Anderson localization using the dynamics of a qubit
Eleuch, H., Hilke, M., & MacKenzie, R. (2017) · Physical Review A · doi:10.1103/physreva.95.062114 · Cited by 8
Higher-order perturbation theory for decoherence in Grover’s algorithm
Azuma, H. (2005) · Physical Review A · doi:10.1103/physreva.72.042305 · Cited by 8
Decoherence-free quantum information in Markovian systems
Patra, M. K., & Brooke, P. G. (2008) · Physical Review A · doi:10.1103/physreva.78.010308 · Cited by 8
Impact of reservoir squeezing on quantum-memory-assisted entropic uncertainty relations
Ali, H., Basit, A., & Li, F. (2025) · Physical Review A · doi:10.1103/physreva.111.032418 · Cited by 7
Transient Synchronization and Quantum Correlations
Paule, G. M. (2018) · Springer theses · doi:10.1007/978-3-319-93964-3_4 · Cited by 7
Internal mechanical dissipation mechanisms in amorphous silicon
Lévesque, C., Roorda, S., Schiettekatte, F., & Mousseau, N. (2022) · Physical Review Materials · doi:10.1103/physrevmaterials.6.123604 · Cited by 7
Dynamics of the nonequilibrium spin-boson model: A benchmark of master equations and their validity
Suárez, G., Łobejko, M., & Horodecki, M. (2024) · Physical Review A · doi:10.1103/physreva.110.042428 · Cited by 7
Effects of imperfect noise correlations on decoherence-free subsystems: SU(2) diffusion model
Demkowicz-Dobrzański, R., Kolenderski, P. L., & Banaszek, K. (2007) · Physical Review A · doi:10.1103/physreva.76.022302 · Cited by 7
Regular patterns in the information flow of local dephasing channels
Giraldi, F. (2017) · Physical Review A · doi:10.1103/physreva.95.022109 · Cited by 8
Adiabatic Transitions in a Two-Level System Coupled to a Free Boson Reservoir
Joye, A., Merkli, M., & Spehner, D. (2020) · Annales Henri Poincaré · doi:10.1007/s00023-020-00946-w · Cited by 7
Universal quantum computation with symmetric qubit clusters coupled to an environment
Boudreault, C., Eleuch, H., Hilke, M., & MacKenzie, R. (2022) · Physical Review A · doi:10.1103/physreva.106.062610 · Cited by 8
Effects of reservoir squeezing on trace-distance correlations and emergence of the pointer basis
Basit, A., Ali, H., Li, F., & Xianlong, G. (2023) · Physical Review A · doi:10.1103/physreva.107.042432 · Cited by 7
Loss of coherence and dressing in QED
Bellomo, B., Compagno, G., & Petruccione, F. (2006) · Physical Review A · doi:10.1103/physreva.74.052112 · Cited by 7
New insights into electron spin dynamics in the presence of correlated noise
Spezia, S., Adorno, D. P., Pizzolato, N., & Spagnolo, B. (2011) · Journal of Physics Condensed Matter · doi:10.1088/0953-8984/24/5/052204 · Cited by 7
Decoherence-free quantum information in the presence of dynamical evolution
Brooke, P. G., Cresser, J. D., & Patra, M. K. (2008) · Physical Review A · doi:10.1103/physreva.77.062313 · Cited by 7
Merkli, M., Berman, G. P., & Redondo, A. (2011) · Journal of Physics A Mathematical and Theoretical · doi:10.1088/1751-8113/44/30/305306 · Cited by 7
Bloch-sphere approach to correlated noise in coupled qubits
Brox, H., Bergli, J., & Galperin, Y. M. (2012) · Journal of Physics A Mathematical and Theoretical · doi:10.1088/1751-8113/45/45/455302 · Cited by 7
Probing polariton dynamics in trapped ions with phase-coherent two-dimensional spectroscopy
Gessner, M., Schlawin, F., & Buchleitner, A. (2015) · The Journal of Chemical Physics · doi:10.1063/1.4919796 · Cited by 7
Fast initial qubit dephasing and the influence of substrate dimensions on error correction rates
Doll, R., Peter Hänggi, A., Kohler, S., & Wubs, M. (2009) · The European Physical Journal B · doi:10.1140/epjb/e2009-00115-6 · Cited by 7
Steganography on quantum pixel images using Shannon entropy
Ortega-Laurel, C., Dong, S., & Cruz‐Irisson, M. (2016) · International Journal of Quantum Information · doi:10.1142/s0219749916500210 · Cited by 7
Production of Entanglement Entropy by Decoherence
Merkli, M., Berman, G. P., Sayre, R. T., Wang, X., & Nesterov, A. I. (2018) · Open Systems & Information Dynamics · doi:10.1142/s1230161218500014 · Cited by 8
The resource theory of coherence for quantum channels
Kamin, F. H., Tabesh, F. T., Salimi, S., & Kheirandish, F. (2020) · Quantum Information Processing · doi:10.1007/s11128-020-02702-9 · Cited by 7
Coherent rovibrational revivals in a thermal molecular ensemble
Bitter, M., Shapiro, E. A., & Milner, V. (2012) · Physical Review A · doi:10.1103/physreva.85.043410 · Cited by 7
Modulating electron dynamics: Modified spin-boson approach
Dahnovsky, Y. (2006) · Physical Review B · doi:10.1103/physrevb.73.144303 · Cited by 7
A Unification of the Coding Theory and OAQEC Perspectives on Hybrid Codes
Majidy, S. (2023) · International Journal of Theoretical Physics · doi:10.1007/s10773-023-05439-0 · Cited by 7
Limitations of practical multiphoton decoherence-free states
Zhang, Y.-S., Li, C., Huang, Y., & Guo, G. (2005) · Physical Review A · doi:10.1103/physreva.72.012308 · Cited by 7
Xu, X., Guo, C., & Chen, R. (2024) · The Journal of Chemical Physics · doi:10.1063/5.0226167 · Cited by 6
Takahashi, H., & Borrelli, R. (2025) · Journal of Chemical Theory and Computation · doi:10.1021/acs.jctc.4c01728 · Cited by 6
Pure phase decoherence in a ring geometry
Zhu, Z. G., Amnon Aharony, A., Ora Entin-wohlman, A., & Stamp, P. C. E. (2010) · Physical Review A · doi:10.1103/physreva.81.062127 · Cited by 6
Segnorile, H. H., & Zamar, R. C. (2013) · The Journal of Chemical Physics · doi:10.1063/1.4824124 · Cited by 6
Connecting two jumplike unravelings for non-Markovian open quantum systems
Luoma, K., Suominen, K., & Piilo, J. (2011) · Physical Review A · doi:10.1103/physreva.84.032113 · Cited by 6
Self-citationReadout of quasiperiodic systems using qubits
Saha, M., Agarwalla, B. K., & Venkatesh, B. P. (2021) · Physical Review A · doi:10.1103/physreva.103.023330 · Cited by 6
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Li, H.-B., Zheng, Q., Zhi, Q., & Li, Y. (2022) · Physical Review A · doi:10.1103/physreva.105.042612 · Cited by 6
On the role of initial coherence in the spin phase-space entropy production rate
Zicari, G., Çakmak, B., Müstecaplıoğlu, Ö. E., & Paternostro, M. (2023) · New Journal of Physics · doi:10.1088/1367-2630/acb45b · Cited by 7
Trushechkin, A. S. (2023) · Mathematics · doi:10.3390/math12010001 · Cited by 6
Pandey, D., & Wubs, M. (2024) · Physical Review Research · doi:10.1103/physrevresearch.6.033044 · Cited by 6
Anders, F. B. (2008) · New Journal of Physics · doi:10.1088/1367-2630/10/11/115007 · Cited by 6
Non-Markovian decay and dynamics of decoherence in private and public environments
Dente, A. D., Zangara, P. R., & Pastawski, H. M. (2011) · Physical Review A · doi:10.1103/physreva.84.042104 · Cited by 6
Entanglement evolution via quantum resonances
Merkli, M. (2011) · Journal of Mathematical Physics · doi:10.1063/1.3637628 · Cited by 6
Initial correlations effects on decoherence at zero temperature
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Proposal for High-Fidelity Quantum Simulation Using a Hybrid Dressed State
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Metadata
- Authors
- Palma, G. M., Suominen, K., & EKERT, A. K.
- Affiliations
- University of Oxford (GB)
- University of Helsinki (FI)
- 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
- 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
- green
- arxiv.org
- 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 › Physics and Astronomy › Atomic and Molecular Physics, and Optics › 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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