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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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Controlling dephasing of coupled qubits via shared bath coherence
Hall, L. M. J., Sirkina, L. S., Morreau, A., Langbein, W., & Muljarov, E. A. (2025) · Physical review. B./Physical review. B · doi:10.1103/ltk8-fpv3 · Cited by 2
Atomic entanglement concentration via photonic Faraday rotation in cavity QED
Xiu, X., Zhao, Z., Wang, X.-Y., Yuan, Z.-Q., Li, J., Ji, Y., & Dong, L. (2024) · Optics Express · doi:10.1364/oe.541186 · Cited by 2
Quantum detection of millimeter wave electric fields with driving surface-state electrons
Zhang, M., Wang, Y. F., Peng, X. Y., Feng, X. N., He, S. R., Li, Y. F., & Wei, L. F. (2025) · Physical Review A · doi:10.1103/pvc5-1hwb · Cited by 2
A unified dynamical framework for quantum sensing based on dephasing interaction
Hu, B., Zhang, Z.-Z., & Wu, W. (2025) · Physics Letters A · doi:10.1016/j.physleta.2025.130944 · Cited by 2
Characterizing correlated noise with single-qubit operations
Gulácsi, B., Kattemölle, J., & Burkard, G. (2025) · Physical Review Research · doi:10.1103/k2lx-vfqp · Cited by 2
Ban, M., Shibata, F., & Kitajima, S. (2007) · Journal of Modern Optics · doi:10.1080/09500340600964114 · Cited by 2
Protection of entanglement between two two-level atoms
Zhang, A. (2017) · Laser Physics Letters · doi:10.1088/1612-202x/aa77c1 · Cited by 2
Model of a programmable quantum processing unit based on a quantum transistor effect
Ablayev, F. M., Andrianov, S. N., Fetisov, D., Moiseev, S. A., Terentyev, A., Urmanchev, A., & Vasiliev, A. (2018) · AIP conference proceedings · doi:10.1063/1.5025452 · Cited by 2
Quantum evolution speed in the finite-temperature bosonic environment
Cheng, J.-Q., Zhang, G., & Xu, J. (2018) · Journal of the Optical Society of America B · doi:10.1364/josab.35.002192 · Cited by 2
Quasi-classical Limit of a Spin Coupled to a Reservoir
Correggi, M., Falconi, M., Fantechi, M., & Merkli, M. (2024) · Quantum · doi:10.22331/q-2024-12-11-1561 · Cited by 2
Frozen and Invariant Quantum Discord Under Local Dephasing Noise
Karpat, G., Addis, C., & Maniscalco, S. (2017) · Quantum science and technology · doi:10.1007/978-3-319-53412-1_16 · Cited by 2
Evolution of quantum coherence induced by band gaps and initial-state preparation measurements
Giraldi, F. (2020) · International Journal of Quantum Information · doi:10.1142/s0219749919410181 · Cited by 2
Utilizing evolution time as a resource to enhance quantum sensing by probe qubit dephasing
Yuan, J.-B., Liu, H.-F., Song, Y.-J., Tang, S.-Q., Wang, X.-W., & Kuang, L.-M. (2025) · Physical Review A · doi:10.1103/74j7-vw6d · Cited by 2
Open Quantum Systems and Decoherence
Suominen, K. (2014) · Springer eBooks · doi:10.1007/978-3-319-04063-9_10 · Cited by 2
Self-citationUniversal quantum logic gates in a scalable Ising spin quantum computer
Mkrtchian, G. F. (2008) · Physics Letters A · doi:10.1016/j.physleta.2008.06.030 · Cited by 2
Critical conditions protecting entanglement
Giraldi, F., & Petruccione, F. (2013) · The European Physical Journal D · doi:10.1140/epjd/e2013-40245-8 · Cited by 2
QUANTUM ERROR CORRECTION DRIVEN ENTANGLEMENT DYNAMICS IN THE PRESENCE OF CORRELATED NOISE
Chiara, G. D., Fazio, R., Macchiavello, C., & Palma, G. M. (2004) · International Journal of Quantum Information · doi:10.1142/s0219749905000761 · Cited by 2
Self-citationPhysical Implementation of Large-Scale Quantum Computation
Suominen, K. (2012) · Springer eBooks · doi:10.1007/978-3-540-92910-9_44 · Cited by 3
Self-citationQuantum information processing in collective-rotating decoherence-free subspace
Su, S., Wang, H., & Zhang, S. (2015) · Quantum Information Processing · doi:10.1007/s11128-015-0975-3 · Cited by 2
Quantum Zeno control of decoherence
Maniscalco, S. (2010) · Laser Physics · doi:10.1134/s1054660x10090094 · Cited by 2
Cheng, J.-Q., Wu, W., & Xu, J. (2017) · Journal of the Optical Society of America B · doi:10.1364/josab.34.000701 · Cited by 2
Dressing of superconducting qubits by their interaction with a low frequency photon reservoir
Gainutdinov, R. K., Nabieva, L. J., Garifullin, A. I., & Mutygullina, A. A. (2019) · Journal of Physics Conference Series · doi:10.1088/1742-6596/1283/1/012004 · Cited by 2
Preparing Decoherence-Recoherence Times of a Qubit with Special Nonselective Measurement Schemes
Giraldi, F. (2021) · Open Systems & Information Dynamics · doi:10.1142/s1230161221500025 · Cited by 3
Persistent nonlocality in an ultracold-atom environment
Longstaff, B., & Brask, J. B. (2023) · Quantum · doi:10.22331/q-2023-01-26-907 · Cited by 2
Quasi-Classical Spin Boson Models
Correggi, M., Falconi, M., & Merkli, M. (2023) · Springer INdAM series · doi:10.1007/978-981-99-5894-8_3 · Cited by 2
Grimaudo, R., MESSINA, A., Nakazato, H., Sergi, A., & Valenti, D. (2023) · Symmetry · doi:10.3390/sym15122174 · Cited by 2
Kinetic State and Emergence of Markovian Dynamics in Exactly Solvable Models of Open Quantum Systems
Trushechkin, A. S. (2024) · Proceedings of the Steklov Institute of Mathematics · doi:10.1134/s0081543824010188 · Cited by 2
Principles of Josephson-Junction-Based Quantum Computation
Shafranjuk, S. E., & Ketterson, J. B. (2008) · Springer eBooks · doi:10.1007/978-3-540-73253-2_9 · Cited by 2
Quantum Computing with Trapped Ions
Lange, W. (2011) · Springer eBooks · doi:10.1007/978-1-4614-1800-9_149 · Cited by 2
Decoherence of encoded quantum registers
Borghoff, S., & Klesse, R. (2008) · Physical Review A · doi:10.1103/physreva.77.012306 · Cited by 2
Bellomo, B., Compagno, G., Nakazato, H., & Yuasa, K. (2009) · Physical Review A · doi:10.1103/physreva.80.052113 · Cited by 2
Decoherence of two-level systems can be very different from Brownian particles
Hu, B. L. (2002) · Chaos Solitons & Fractals · doi:10.1016/s0960-0779(02)00299-0 · Cited by 3
Environment and initial state engineered dynamics of quantum and classical correlations
Wang, C., Li, C.-X., Guo, Y. J., Lu, G., & Ding, K.-H. (2016) · Annals of Physics · doi:10.1016/j.aop.2016.08.014 · Cited by 2
Resonance Dynamics and Decoherence
Merkli, M. (2009) · Birkhäuser Basel eBooks · doi:10.1007/978-3-7643-9906-1_20 · Cited by 2
Martín-Vázquez, G., Gert Aarts, A., Müller, M., & Bermúdez, A. (2022) · PRX Quantum · doi:10.1103/prxquantum.3.020352 · Cited by 5
Sensing microscopic noise events by frequent quantum measurements
Virzì, S., Knoll, L. T., Avella, A., Piacentini, F., Gherardini, S., Gramegna, M., Kurizki, G., Kofman, A. G., Degiovanni, I. P., Genovese, M., et al. (2024) · Physical Review Applied · doi:10.1103/physrevapplied.21.034014 · Cited by 2
Quantum Error Correction Is Applicable for Reducing Spatially Correlated Decoherence
Duan, L.-M., & Guo, G. (1999) · Lecture notes in computer science · doi:10.1007/3-540-49208-9_30 · Cited by 2
Targeting qubit states using open-loop control
Helon, C. D., Protopopescu, V., & Perez, R. (2003) · Journal of Physics A Mathematical and General · doi:10.1088/0305-4470/36/25/315 · Cited by 2
Quantum coding and decoding in multi-level adiabatic passage schemes
Král, P., & Shapiro, M. (2004) · Chemical Physics Letters · doi:10.1016/j.cplett.2004.06.091 · Cited by 2
1/F Noise During Manipulation of Josephson Charge Qubits
Paladino, E., Faoro, L., Falci, G., & Fazio, R. (2001) · Springer eBooks · doi:10.1007/978-1-4615-1245-5_36 · Cited by 2
Computational spin dynamics and visualisation of large spin systems
Charnock, G. T. P., & Charnock, G. (2012) · Oxford University Research Archive (ORA) (University of Oxford) · doi:10.5287/ora-8rx1vwkpb · Cited by 2
Quantum Key Distribution: Security, Feasibility and Robustness
Wang, X. (2006) · Springer eBooks · doi:10.1007/11683858_8 · Cited by 2
Decoherence of Two-Level Systems Can Be Very Different From Brownian ParticlesPreprint
Hu, B. L. (2002) · Cited by 2
Mathews, M. M., & Panchami, V. (2021) · SSRN Electronic Journal · doi:10.2139/ssrn.3882506 · Cited by 2
Real time dynamics and critical phenomena of quantum impurity systems
Güttge, F. (2013) · Technische Universität Dortmund Eldorado (Technische Universität Dortmund) · doi:10.17877/de290r-6954 · Cited by 2
Improving cutoff frequency estimation via optimized π-pulse sequence
Zheng, W.-S., Zhang, C., & Gong, B. (2024) · Chinese Physics B · doi:10.1088/1674-1056/ad9ba1 · Cited by 1
Barrada, K. (2025) · Axioms · doi:10.3390/axioms14110844 · Cited by 1
A numerically exact, non-Markovian, non-Gaussian noise model for open quantum system dynamics
Lin, Z., & Cheng, Y. (2026) · The Journal of Chemical Physics · doi:10.1063/5.0331181 · Cited by 1
Fault Tolerant Practical Quantum DigitalSignature against Collective Noises
Wu, J., Yue, L.-Y., XIE, J., Liu, X., & Zhang, M. (2021) · Optics Continuum · doi:10.1364/optcon.446301 · Cited by 1
Computable and Faithful Lower Bound on Entanglement Cost
Wang, X., Jing, M., & Zhu, C. (2025) · Physical Review Letters · doi:10.1103/physrevlett.134.190202 · Cited by 4
Theoretical Tools for Quantum Optics in Structured Media
Mogilevtsev, D. S., & Kilin, S. Y. (2009) · Progress in optics · doi:10.1016/s0079-6638(10)05407-7 · Cited by 1
FIDELITY AND ENTANGLEMENT OF A SPATIALLY EXTENDED LINEAR THREE-QUBIT REGISTER
Doll, R., Wubs, M., Kohler, S., & Peter Hänggi, A. (2008) · International Journal of Quantum Information · doi:10.1142/s0219749908003955 · Cited by 1
The initial condition problem of damped quantum harmonic oscillator
Gao, Y., O’Connell, R. F., Tang, Q. B., & Wang, R. M. (2015) · The European Physical Journal D · doi:10.1140/epjd/e2014-50242-0 · Cited by 1
SOME APPLICATIONS OF THE BURSHTEIN EQUATION TO A FIELD-NOISY JAYNES–CUMMINGS INTERACTIONS
Kayhan, H. (2011) · International Journal of Quantum Information · doi:10.1142/s021974991100799x · Cited by 1
From Local Operations to Collective Dephasing: Behavior of Correlated Quantum States
Gessner, M. (2016) · Springer theses · doi:10.1007/978-3-319-44459-8_3 · Cited by 1
Non-classicality dynamics of Schrödinger cat states in a non-Markovian environment
Cheng, X., Xiang, S.-H., & Zhao, Y.-J. (2020) · Physica Scripta · doi:10.1088/1402-4896/ab8c67 · Cited by 1
Paladino, E., Faoro, L., Falci, G., & Fazio, R. (2002) · Springer eBooks · doi:10.1007/978-1-4615-0737-6_3 · Cited by 1
Su, H., Li, J., Xu, Y., Qi, D., Zhang, S.-F., Zhang, L.-L., & Gong, W. (2023) · Chinese Journal of Physics · doi:10.1016/j.cjph.2023.11.011 · Cited by 1
Quantized Approach to Damped Transversal Mechanical Waves
Márkus, F., & Gambár, K. (2024) · Quantum Reports · doi:10.3390/quantum6010009 · Cited by 1
Said, A. K., Javed, M., & Ali, Z. (2024) · International Journal of Theoretical Physics · doi:10.1007/s10773-024-05838-x · Cited by 1
Quantum logic operations on two distant atoms trapped in two optical-fibre-connected cavities
Zhang, Y.-Q., Zhang, S., Yeon, K.-H., & Yu, S. (2011) · Chinese Physics B · doi:10.1088/1674-1056/20/12/120310 · Cited by 1
Dynamical stability of entanglement between spin ensembles
Ng, H. T., & Bose, S. (2010) · Journal of Physics A Mathematical and Theoretical · doi:10.1088/1751-8113/43/7/072001 · Cited by 1
Two-Photon Jaynes–Cummings Model Governed by Milburn Equation with Phase Damping
Chen, C., Li, S., & Zong-Liang, L. (2006) · Communications in Theoretical Physics · doi:10.1088/0253-6102/45/4/026 · Cited by 1
Short-time decoherence of a quantum Hadamard gate with Josephson charge qubits
Li, J., Li, Z.-J., Gao, Y., & Liang, J. (2006) · Physica B Condensed Matter · doi:10.1016/j.physb.2005.09.029 · Cited by 1
Environment-Induced Entanglement for a Tripartite System in Non-Markovian Dissipation
Li, Y., Huang, J., & Xiao, X. M. (2013) · International Journal of Theoretical Physics · doi:10.1007/s10773-013-1685-1 · Cited by 1
Quantum beat and entanglement of multi-qubits interacting with a common reservoir
Sato, A., Ishi‐Hayase, J., Minami, F., & Sasaki, M. (2006) · Journal of Luminescence · doi:10.1016/j.jlumin.2006.01.044 · Cited by 1
Burgarth, D. K., & Giovannetti, V. (2013) · Springer eBooks · doi:10.1007/978-3-642-39937-4_3 · Cited by 1
Prefect Protection of Quantum-Enhanced Metrology from Dephasing Noise*
Zhong, W., Huang, Y., & Wang, X. (2017) · Communications in Theoretical Physics · doi:10.1088/0253-6102/67/4/383 · Cited by 1
Giraldi, F. (2018) · Open Systems & Information Dynamics · doi:10.1142/s1230161218500117 · Cited by 1
Protecting quantum systems from decoherence with unitary operations
Kiktenko, E. O., Mastiukova, A. S., & Fedorov, A. K. (2020) · Optical Engineering · doi:10.1117/1.oe.59.6.061625 · Cited by 1
Degenerate Landau–Zener model in the presence of quantum noise
Militello, B. (2019) · International Journal of Quantum Information · doi:10.1142/s0219749919500497 · Cited by 1
Decoding Quantum Error Correction Codes With Local Variation
Hanks, M., Munro, W. J., & Nemoto, K. (2020) · IEEE Transactions on Quantum Engineering · doi:10.1109/tqe.2020.2967890 · Cited by 1
Segnorile, H. H., González, C. E., & Zamar, R. C. (2021) · Annals of Physics · doi:10.1016/j.aop.2021.168655 · Cited by 1
Robustness of deformed catlike states under dissipative decoherence
Belfakir, A., Belhaj, A., & Hassouni, Y. (2020) · Physical review. D/Physical review. D. · doi:10.1103/physrevd.102.065003 · Cited by 1
UNDAMENTALS OF INFORMATION AND COMPUTATION IN THE REALM OF THE QUANTA
Reina, J. H. (2023) · Revista de la Academia Colombiana de Ciencias Exactas Físicas y Naturales · doi:10.18257/raccefyn.33(127).2009.2387 · Cited by 1
Quantum Computing: A Way to Break Complexity?
Gesù, V. D., & Palma, G. M. (2003) · Springer eBooks · doi:10.1007/978-94-010-0155-7_21 · Cited by 1
Self-citationDynamic Decoupling Quantum Control Methods
(2014) · Wiley eBooks · doi:10.1002/9781118608135.ch14 · Cited by 1
Desacoplamento dinâmico de estados quânticos via campos contínuos de alta frequência
Fanchini, F. F. (2008) · Digital Library of Theses and Dissertations (Universidade de São Paulo) · doi:10.11606/t.76.2008.tde-19052011-090931 · Cited by 1
Photonic Realization of Quantum Information Protocols
Genovese, M. (2010) · Lecture notes in physics · doi:10.1007/978-3-642-11914-9_7 · Cited by 1
Collective quantum decoherence induced by qubit–electron interaction
Gao, Y., & Xiong, S. (2008) · Physics Letters A · doi:10.1016/j.physleta.2008.04.028 · Cited by 1
Cavity loss induced generation of W states
Wu, H., Yang, Z., Su, W.-J., Zhi-Rong, Z., & Shi-Biao, Z. (2008) · Chinese Physics B · doi:10.1088/1674-1056/17/1/009 · Cited by 1
Molecular architectures based on π-conjugated block copolymers for global quantum computation
Mujica‐Martinez, C. A., Arce, J. C., Reina, J. H., & Thorwart, M. (2009) · Journal of Physics Conference Series · doi:10.1088/1742-6596/167/1/012061 · Cited by 1
Imitating quantum mechanics: Qubit-based model for simulation
Peil, S. (2009) · Physical Review A · doi:10.1103/physreva.79.042320 · Cited by 1
Encoded Universality of Quantum Computations on the Multi-Atomic Ensembles in the QED CavityPreprint
Ablayev, F. M., Andrianov, S. N., Moiseev, S. A., & Vasiliev, A. (2011) · arXiv (Cornell University) · doi:10.48550/arxiv.1109.0291 · Cited by 1
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Quantum Information Processing: A Brief Overview of Recent Advances
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Optimized Dynamical Decoupling in Ξ-Type n-Level Quantum SystemsPreprint
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Chapter 24 Non-Hamiltonian Systems as Quantum Computers
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Purity preservation of quantum systems by resonant field
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Open n-Qubit System as a Quantum Computer with Four-Valued LogicPreprint
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Quantum Error-Correcting Code for Burst Error
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Superposition, Entanglement and Other Oddities
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Physical challenges of quantum computation
Yung, M. H. (2009) · Illinois Digital Environment for Access to Learning and Scholarship (University of Illinois at Urbana-Champaign)
Classical manipulation of a quantum system
Mehmani, B. (2010) · UvA-DARE (University of Amsterdam)
The amplified spontaneous emission in TE CO2 laser with a half-cavity
Mei, C., Li, Y., Jinglun, L., & Junping, G. (2007) · Optics & Laser Technology · doi:10.1016/j.optlastec.2007.10.001
ENTROPY GROWTH INDUCED BY A SQUEEZED FIELD WITH A PHASE-DAMPING RESERVOIR
Obada, A. ‐. F., Hessian, H. A., & Mohamed, A. ‐. A. (2009) · International Journal of Modern Physics B · doi:10.1142/s0217979209053205
Cavity-induced quantum cooperative phenomena
Plastina, F., Falcone, G., Francica, F., Liberti, G., Piperno, F., & Maniscalco, S. (2010) · Physica Scripta · doi:10.1088/0031-8949/2010/t140/014008
Azuma, H. (2005) · Physical Review A · doi:10.1103/physreva.72.049902
CONTROL AND ERROR PREVENTION IN CONDENSED MATTER QUANTUM COMPUTING DEVICES
Byrd, M., & Wu, L.-A. (2007) · International Journal of Modern Physics B · doi:10.1142/s0217979207043841
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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