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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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Items 401–500 of 878
One-way quantum computing in a decoherence-free subspace
Tame, M., Paternostro, M., & Kim, M. S. (2007) · New Journal of Physics · doi:10.1088/1367-2630/9/6/201 · Cited by 17
Wang, S.-J., An, J., Luo, H., & Jia, C. (2003) · Journal of Physics A Mathematical and General · doi:10.1088/0305-4470/36/3/316 · Cited by 17
Electromagnetic fluctuations near thin metallic films
Langsjoen, L. S., Poudel, A., Vavilov, M., & Joynt, R. (2014) · Physical Review B · doi:10.1103/physrevb.89.115401 · Cited by 16
Semi-quantum secure direct communication against collective-dephasing noise
Wang, P., Chen, X., & Sun, Z. (2022) · Quantum Information Processing · doi:10.1007/s11128-022-03702-7 · Cited by 16
Nourmandipour, A., & Tavassoly, M. K. (2015) · The European Physical Journal Plus · doi:10.1140/epjp/i2015-15148-7 · Cited by 16
System-environment correlations for dephasing two-qubit states coupled to thermal baths
Costa, A. C. S., Beims, M. W., & Strunz, W. T. (2016) · Physical Review A · doi:10.1103/physreva.93.052316 · Cited by 16
Quantum coherence of spin-boson model at finite temperature
Wu, W., & Xu, J. (2017) · Annals of Physics · doi:10.1016/j.aop.2017.01.014 · Cited by 17
Quantum error avoiding codes verses quantum error correcting codes
Duan, L.-M., & Guo, G. (1999) · Physics Letters A · doi:10.1016/s0375-9601(99)00183-8 · Cited by 16
Active suppression of dephasing in Josephson-junction qubits
Averin, D. V., & Fazio, R. (2003) · Journal of Experimental and Theoretical Physics Letters · doi:10.1134/1.1644314 · Cited by 16
Abrupt decay of entanglement and quantum communication through noise channels
METWALLY, N. (2009) · Quantum Information Processing · doi:10.1007/s11128-009-0149-2 · Cited by 16
Theory of multiwave mixing and decoherence control in a qubit array system
Sasaki, M., Hasegawa, A., Ishi‐Hayase, J., Mitsumori, Y., & Minami, F. (2005) · Physical Review B · doi:10.1103/physrevb.71.165314 · Cited by 16
Decoherence suppression of excitons in GaSe using three successive femtosecond optical pulses
Kishimoto, T. E., Hasegawa, A., Mitsumori, Y., Ishi‐Hayase, J., Sasaki, M., & Minami, F. (2006) · Physical Review B · doi:10.1103/physrevb.74.073202 · Cited by 16
Fast incomplete decoherence of nuclear spins in a quantum Hall ferromagnet
Maniv, T., Bychkov, Y. A., Vagner, I. D., & Wyder, P. (2001) · Physical review. B, Condensed matter · doi:10.1103/physrevb.64.193306 · Cited by 16
Collective couplings: Rectification and supertransmittance
Schaller, G., Giusteri, G. G., & Celardo, G. L. (2016) · Physical review. E · doi:10.1103/physreve.94.032135 · Cited by 16
Time-invariant discord in dynamically decoupled systems
Addis, C., Karpat, G., & Maniscalco, S. (2015) · Physical Review A · doi:10.1103/physreva.92.062109 · Cited by 15
Survival of coherence for open quantum systems in thermal baths
Giraldi, F., & Petruccione, F. (2013) · Physical Review A · doi:10.1103/physreva.88.042102 · Cited by 15
Zhu, X., Jin, Z., Liang, E., Zhang, S., & Su, S. (2020) · Annalen der Physik · doi:10.1002/andp.202000059 · Cited by 15
Passively self-error-rejecting qubit transmission over a collective-noise channel
Deng, F., Li, X., & Zhou, H.-Y. (2011) · Quantum Information and Computation · doi:10.26421/qic11.11-12-2 · Cited by 15
Mishima, K., & Yamashita, K. (2009) · The Journal of Chemical Physics · doi:10.1063/1.3159002 · Cited by 15
Problem of coherent control in non-Markovian open quantum systems
Addis, C., Laine, E.-M., Gneiting, C., & Maniscalco, S. (2016) · Physical Review A · doi:10.1103/physreva.94.052117 · Cited by 15
Decoherence-Free Rotational Degrees of Freedom for Quantum Applications
Pedernales, J. S., Cosco, F., & Plenio, M. B. (2020) · Physical Review Letters · doi:10.1103/physrevlett.125.090501 · Cited by 15
Quantum metrology with one auxiliary particle in a correlated bath and its quantum simulation
He, W., Guang, H.-Y., Li, Z., Deng, R.-Q., Zhang, N., Zhao, J.-X., Deng, F., & Ai, Q. (2021) · Physical Review A · doi:10.1103/physreva.104.062429 · Cited by 15
Environment-assisted analog quantum search
Novo, L., Chakraborty, S., Mohseni, M., & Omar, Y. (2018) · Physical Review A · doi:10.1103/physreva.98.022316 · Cited by 16
Quantum metrology out of equilibrium
Razavian, S., & Paris, M. G. A. (2019) · Physica A Statistical Mechanics and its Applications · doi:10.1016/j.physa.2019.03.125 · Cited by 16
Wang, F., Huang, Y., Zhang, Z., Zu, C., Hou, P., Yuan, X., Wang, W.-B., Zhang, W.-G., He, L., Chang, X.-Y., et al. (2017) · Physical review. B./Physical review. B · doi:10.1103/physrevb.96.134314 · Cited by 14
Fedichkin, L. E., & Fedorov, A. (2005) · IEEE Transactions on Nanotechnology · doi:10.1109/tnano.2004.840156 · Cited by 14
Generation of four-photon polarization-entangled decoherence-free states within a network
Xia, Y., Song, J., Yang, Z., & Zheng, S. B. (2010) · Applied Physics B · doi:10.1007/s00340-010-3937-6 · Cited by 14
Scale-invariant freezing of entanglement
Chanda, T., Das, T., Sadhukhan, D., Pal, A. K., Sen, A., & Sen, U. (2018) · Physical Review A · doi:10.1103/physreva.97.062324 · Cited by 14
Effective description of the short-time dynamics in open quantum systems
Rossi, M. A. C., Foti, C., Cuccoli, A., Trapani, J., Verrucchi, P., & Paris, M. G. A. (2017) · Physical Review A · doi:10.1103/physreva.96.032116 · Cited by 14
Realization of hierarchical equations of motion from stochastic perspectives
Wu, W. (2018) · Physical Review A · doi:10.1103/physreva.98.012110 · Cited by 14
Agents, Subsystems, and the Conservation of Information
Chiribella, G. (2018) · Entropy · doi:10.3390/e20050358 · Cited by 19
Concatenating dynamical decoupling with decoherence-free subspaces for quantum computation
Zhang, Y., Zhou, Z.-W., Yu, B. H., & Guo, G. (2004) · Physical Review A · doi:10.1103/physreva.69.042315 · Cited by 15
Robust implementations of quantum repeaters
Klein, A., Dorner, U., Alves, C. M., & Jaksch, D. (2006) · Physical Review A · doi:10.1103/physreva.73.012332 · Cited by 14
Saving entanglement via a nonuniform sequence of π pulses
Agarwal, G. S. (2010) · Physica Scripta · doi:10.1088/0031-8949/82/03/038103 · Cited by 14
Floquet stroboscopic divisibility in non-Markovian dynamics
Bastidas, V. M., Kyaw, T. H., Tangpanitanon, J., Romero, G., Kwek, L.-C., & Angelakis, D. G. (2018) · New Journal of Physics · doi:10.1088/1367-2630/aadcbd · Cited by 13
Fault-Tolerate Three-Party Quantum Secret Sharing over a Collective-Noise Channel
Li, C., & Li, Y. (2011) · Chinese Physics Letters · doi:10.1088/0256-307x/28/2/020304 · Cited by 13
Material science for quantum computing with atom chips
Folman, R. (2011) · Quantum Information Processing · doi:10.1007/s11128-011-0311-5 · Cited by 13
Mean-field evolution of open quantum systems: an exactly solvable model
Merkli, M., & Berman, G. P. (2012) · Proceedings of the Royal Society A Mathematical Physical and Engineering Sciences · doi:10.1098/rspa.2012.0327 · Cited by 13
Generating and protecting correlated quantum states under collective dephasing
Carnio, E. G., Buchleitner, A., & Gessner, M. (2016) · New Journal of Physics · doi:10.1088/1367-2630/18/7/073010 · Cited by 15
One-way Transfer of Quantum States via Decoherence
Matsuzaki, Y., Bastidas, V. M., Takeuchi, Y., Munro, W. J., & Saito, S. (2020) · Journal of the Physical Society of Japan · doi:10.7566/jpsj.89.044003 · Cited by 13
Appearance and stability of anomalously fluctuating states in Shor’s factoring algorithm
Ukena, A., & Shimizu, A. (2004) · Physical Review A · doi:10.1103/physreva.69.022301 · Cited by 13
Hao, Y., Song, J., Qin, H., Han, L.-F., Hou, K., Xiao-Yuan, H., & Shou-Hua, S. (2008) · Communications in Theoretical Physics · doi:10.1088/0253-6102/50/3/19 · Cited by 13
Generalized multiplicative domains and quantum error correction
Johnston, N., & Kribs, D. W. (2010) · Proceedings of the American Mathematical Society · doi:10.1090/s0002-9939-2010-10556-7 · Cited by 14
Quantum measurements without macroscopic superpositions
Spehner, D., & Haake, F. (2008) · Physical Review A · doi:10.1103/physreva.77.052114 · Cited by 13
Engineering selective linear and nonlinear Jaynes–Cummings interactions and applications
Prado, F. O., Rosado, W., Alcalde, A. M., & Moussa, M. H. Y. (2013) · Journal of Physics B Atomic Molecular and Optical Physics · doi:10.1088/0953-4075/46/20/205501 · Cited by 13
Communication between general-relativistic observers without a shared reference frame
Chęcińska, A., & Dragan, A. (2015) · Physical Review A · doi:10.1103/physreva.92.012321 · Cited by 12
State protection under collective damping and diffusion
de Ponte, M. A., Mizrahi, S. S., & Moussa, M. H. Y. (2011) · Physical Review A · doi:10.1103/physreva.84.012331 · Cited by 12
Extreme depolarization for any spin
Denis, J., & Martin, J. (2022) · Physical Review Research · doi:10.1103/physrevresearch.4.013178 · Cited by 12
Unraveling metastable Markovian open quantum systems
Brown, C. A., Macieszczak, K., & Jack, R. L. (2024) · Physical Review A · doi:10.1103/physreva.109.022244 · Cited by 15
Suppression of decoherence in quantum registers by entanglement with a nonequilibrium environment
Gheorghiu-Svirschevski, S. (2002) · Physical Review A · doi:10.1103/physreva.66.032101 · Cited by 11
Fixed points of commutative super-operators
Long, L., & Zhang, S. (2011) · Journal of Physics A Mathematical and Theoretical · doi:10.1088/1751-8113/44/9/095201 · Cited by 12
Effect of bath temperature on the decoherence of quantum dissipative systems
Wu, W., & Lin, H.-Q. (2016) · Physical Review A · doi:10.1103/physreva.94.062116 · Cited by 12
Modulation of dephasing due to a spin-boson environment
Paladino, E., Sassetti, M., & Falci, G. (2003) · Chemical Physics · doi:10.1016/j.chemphys.2003.08.027 · Cited by 12
Double-dot chain as a macroscopic quantum bit
PASQUALE, F. D., Giorgi, G. L., & Paganelli, S. (2005) · Physical Review A · doi:10.1103/physreva.71.042304 · Cited by 12
Ji, Y., & Hu, J. (2019) · Physica E Low-dimensional Systems and Nanostructures · doi:10.1016/j.physe.2019.113583 · Cited by 12
MEASUREMENT OF A QUANTUM SYSTEM COUPLED TO INDEPENDENT HEAT-BATH AND POINTER MODES
MOZYRSKY, D., & Vladimir Privman, A. (2000) · Modern Physics Letters B · doi:10.1142/s0217984900000409 · Cited by 16
Effects of Imperfect Gate Operations in Shor's Prime Factorization Algorithm
Guo, H., Long, G., & Sun, Y. (2001) · Journal of the Chinese Chemical Society · doi:10.1002/jccs.200100067 · Cited by 13
Two cavity modes in a dissipative environment: Cross decay rates and robust states
Rossi, R., de Magalhães, A. R. B., & Nemes, M. C. (2005) · Physica A Statistical Mechanics and its Applications · doi:10.1016/j.physa.2005.09.072 · Cited by 12
Effective Physical Processes and Active Information in Quantum ComputingPreprint
Licata, I. F. (2007) · arXiv (Cornell University) · doi:10.48550/arxiv.0705.1173 · Cited by 12
Dynamically generated decoherence-free subspaces and subsystems on superconducting qubits
Quiroz, G., Pokharel, B., Boen, J., Tewala, L., Tripathi, V., Williams, D., Wu, L.-A., Titum, P., Schultz, K., & Lidar, D. A. (2024) · Reports on Progress in Physics · doi:10.1088/1361-6633/ad6805 · Cited by 15
Effective modeling of open quantum systems by low-rank discretization of structured environments
Takahashi, H., & Borrelli, R. (2024) · The Journal of Chemical Physics · doi:10.1063/5.0232232 · Cited by 11
Unconventional solitonic high-temperature superfluorescence from perovskites
Biliroglu, M., Türe, M., Ghita, A., Kotyrov, M., Qin, X., Seyitliyev, D., Phonthiptokun, N., Abdelsamei, M., Chai, J., Su, R., et al. (2025) · Nature · doi:10.1038/s41586-025-09030-x · Cited by 15
Entropy production in open quantum systems: exactly solvable qubit models
Morozov, & RÖPKE, G. (2012) · Condensed Matter Physics · doi:10.5488/cmp.15.43004 · Cited by 11
Amount of quantum coherence needed for measurement incompatibility
Kiukas, J., McNulty, D., & Pellonpää, J.-P. (2022) · Physical Review A · doi:10.1103/physreva.105.012205 · Cited by 11
Frequency estimation under non-Markovian spatially correlated quantum noise
Riberi, F., Norris, L., Beaudoin, F., & Viola, L. (2022) · New Journal of Physics · doi:10.1088/1367-2630/ac92a2 · Cited by 13
Non-Markovian incoherent quantum dynamics of a two-state system
Amin, M. H. S., & Brito, F. (2009) · Physical Review B · doi:10.1103/physrevb.80.214302 · Cited by 11
The dynamical role of initial correlation in the exactly solvable dephasing model
Gao, Y. (2013) · The European Physical Journal D · doi:10.1140/epjd/e2013-40158-6 · Cited by 11
Memory effects in spontaneous emission processes
Grimsmo, A. L., Vaskinn, A. H., Rekdal, P. K., & Skagerstam, B.-S. (2013) · Physical Review A · doi:10.1103/physreva.87.022101 · Cited by 11
Realization of universal nonadiabatic geometric control on decoherence-free qubits in the XY model
Mousolou, V. A. (2018) · Europhysics Letters (EPL) · doi:10.1209/0295-5075/121/20004 · Cited by 11
Improved entanglement detection with subspace witnesses
Sun, W. K. C., Cooper, A., & Cappellaro, P. (2020) · Physical Review A · doi:10.1103/physreva.101.012319 · Cited by 11
Martin, T., & Tene, A. G. (2020) · International Journal of Theoretical Physics · doi:10.1007/s10773-020-04502-4 · Cited by 11
Quantum information and triangular optical lattices
Kay, A., Lee, D. K. K., Pachos, J. K., Plenio, M. B., Reuter, M. E., & Rico, E. (2005) · Optics and Spectroscopy · doi:10.1134/1.2055926 · Cited by 12
Dephasing of qubits by the Schrödinger cat
Dajka, J., Mierzejewski, M., Łuczka, J., & Peter Hänggi, A. (2009) · Physica E Low-dimensional Systems and Nanostructures · doi:10.1016/j.physe.2009.06.080 · Cited by 11
Entanglement of unstable atoms: modifications of the emission properties
Sancho, P., & Plaja, L. (2009) · Journal of Physics B Atomic Molecular and Optical Physics · doi:10.1088/0953-4075/42/16/165008 · Cited by 11
Electron transport across a quantum wire in the presence of electron leakage to a substrate
Kwapiński, T., Kohler, S., & Peter Hänggi, A. (2010) · The European Physical Journal B · doi:10.1140/epjb/e2010-10452-x · Cited by 11
Facchi, P., Marmo, G., Pascazio, S., Scardicchio, A., & Sudarshan, E. C. G. (2004) · Journal of Optics B Quantum and Semiclassical Optics · doi:10.1088/1464-4266/6/6/006 · Cited by 11
Virtual processes and super-radiance in spin-boson models
Alcalde, M. A., Kullock, R., & Svaiter, N. F. (2009) · Journal of Mathematical Physics · doi:10.1063/1.3040187 · Cited by 11
Scaling of decoherence effects in quantum computers
Dalton, B. J. (2003) · Journal of Modern Optics · doi:10.1080/09500340308234544 · Cited by 11
Collective decoherence of the superpositional entangled states in the quantum Shor algorithm
Berman, G. P., Kamenev, D. I., & Tsifrinovich, V. I. (2005) · Physical Review A · doi:10.1103/physreva.71.032346 · Cited by 12
Effects of dynamical phases in Shor’s factoring algorithm with operational delays
Wei, L. F., Li, X., Hu, X., & Nori, F. (2005) · Physical Review A · doi:10.1103/physreva.71.022317 · Cited by 11
A small trapped-ion quantum register
Kielpinski, D. (2003) · Journal of Optics B Quantum and Semiclassical Optics · doi:10.1088/1464-4266/5/3/201 · Cited by 11
Coherence in a network of two-level systems coupled to a bosonic field
Fischer, S., & Breuer, H. (2013) · Physical Review A · doi:10.1103/physreva.88.062103 · Cited by 10
Quantum error mitigation via quantum-noise-effect circuit groups
Hama, Y., & Nishi, H. (2024) · Scientific Reports · doi:10.1038/s41598-024-52485-7 · Cited by 10
Sub-exponential spin-boson decoherence in a finite bath
Wong, V., & Gruebele, M. (2002) · Chemical Physics · doi:10.1016/s0301-0104(02)00534-7 · Cited by 10
Microscopic quantum dynamics study on the noise threshold of fault-tolerant quantum error correction
Cheng, Y., & Silbey, R. (2005) · Physical Review A · doi:10.1103/physreva.72.012320 · Cited by 10
Engineered noisy environment for studying decoherence
岩倉, 愛., Matsuzaki, Y., & Kondo, Y. (2017) · Physical Review A · doi:10.1103/physreva.96.032303 · Cited by 10
Microscopic biasing of discrete-time quantum trajectories
Cilluffo, D., Buonaiuto, G., Lesanovsky, I., Carollo, A. C. M., Lorenzo, S., Palma, G. M., Ciccarello, F., & Carollo, F. (2021) · Quantum Science and Technology · doi:10.1088/2058-9565/ac15e2 · Cited by 12
Self-citationAspects of quantum coherence in the optical Bloch equations
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Entanglement sudden death: a threat to advanced quantum key distribution?
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Geometric entangling gates for coupled cavity system in decoherence-free subspaces
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Decoherence Dynamics of a Superconducting Charge Qubit Coupled to a Boson Bath and a Spin Bath
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Complete experimental toolbox for alignment-free quantum communication
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Bell-state preparation for electron spins in a semiconductor double quantum dot
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Quantum error correction with spatially correlated decoherence
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Conditions for independent decoherence
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Robust quantum searching with spontaneously decaying qubits
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Entanglement evolution via quantum resonancesPreprint
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Spacetime foam induced collective bundling of intense fields
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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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