Internal consistency of fault-tolerant quantum error correction in light of rigorous derivations of the quantum Markovian limit

In the weak coupling limit, a new master equation for a system subject to periodic driving is derived.

“As part of our discussion we derive, in the weak coupling limit, a new master equation for a system subject to periodic driving.”

For the ideal QEC recovery, the fidelity F equals 1 if and only if the ancilla state is pure (ρ_A = |0_A><0_A|).

“Clearly, F = 1 iff ρ A = | 0 A 〉〈 0 A | , i.e., the ancillas are pure .”

The conclusion that the ancillas' state must be pure is unchanged when non-unitary recovery via ancilla measurements and conditional unitaries is used.

“The conclusion that the ancillas' state must be pure is unchanged.”

The KMS condition implies a strong asymmetry of the spectral density G(ω) for low T, where T is measured relative to the presence of kT energy scales in the bath.

“The KMS condition implies a strong asymmetry of the spectral density G ( ω ) for low T , where T is measured relative to the presence of kT energy scales in the bath, i.e., relative to the range where G ( ω ) is non-vanishing.”

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• AssumptionThe authors state this as an assumption: The reservoir (not the quantum computer) is in thermal equilibrium.

The Markovian condition can hold only if kT ≫ ω over the entire spectrum of the system's Bohr frequencies.

“More precisely, the Markovian condition can hold only if kT ≫ ω over the entire spectrum of the system's Bohr frequencies .”

• AssumptionThe authors state this as an assumption: The reservoir (not the quantum computer) is in thermal equilibrium.

The weak coupling limit is inconsistent with fast gates.

“while the latter is inconsistent with fast gates.”

The singular coupling limit physically makes sense only in the high-temperature limit.

“It is based on a rescaling of the bath and system-bath Hamiltonians, which physically makes sense in the hightemperature limit only.”

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The requirement of a constant supply of nearly pure and cold ancillas contradicts the high-temperature limit needed for the Markov approximation to hold.

“the issue being essentially that the requirement of a constant supply of nearly pure and cold ancillas contradicts the highT limit needed for the Markov approximation to hold.”

The Davies derivation can be generalized to allow for slowly varying system Hamiltonians, whenever the time scale of variation of H_C(t) is much longer than the inverse of the typical Bohr frequency of H_S.

“Davies did not consider time-dependent system Hamiltonians in [43], but it is possible to generalize his derivation to allow for slowly varying system Hamiltonians [37, 44, 45]. That is, whenever the time scale of the variation of H C ( t ) is much longer than the inverse of the typical Bohr frequency (of H S ), it is possible to add H C ( t ) to the system Hamiltonian in Eq. (23), necessitating a”

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The Davies weak-coupling derivation imposes severe restrictions on the speed of quantum gates.

“Hence, while the Davies approach does not require the highT limit, it imposes severe restrictions on the speed of quantum gates.”

In the Davies approach, the Markov approximation is a consequence of weak coupling and time coarse-graining, which leads to cancellation of non-Markovian oscillating terms.

“In the Davies approach the Markov approximation is a consequence of weak coupling (and hence slow dynamics of the system in the interaction picture), and time coarse-graining, which leads to cancellation of the non-Markovian oscillating terms.”

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The singular coupling limit is incompatible with the assumption of a constant and fresh supply of cold ancillas.

“The former is incompatible with the assumption of a constant and fresh supply of cold ancillas”

The Davies weak-coupling derivation is invalid for time-dependent Hamiltonians, except in the adiabatic case.

“The price we pay is the invalidity of this approach for timedependent Hamiltonians, except in the adiabatic case.”

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Markovian dynamics requires either the singular coupling limit (high temperature) or the weak coupling limit (weak system-bath interaction).

“Namely, Markovian dynamics requires either the singular coupling limit (high temperature), or the weak coupling limit (weak system-bath interaction).”

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The Markovian approximation can be consistent with the KMS condition only in the high-temperature regime kT ≫ E, where E is the system energy scale.

“this leads to the important realization that the Markovian approximation can be consistent with the KMS condition only in the high temperature regime kT ≫ E , where E is the system energy scale .”

• AssumptionThe authors state this as an assumption: The reservoir (not the quantum computer) is in thermal equilibrium.

A flat spectral density is possible only in the limit T→∞ of the KMS condition.

“However, this is possible only in the limit T → ∞ of the KMS condition.”

• AssumptionThe authors state this as an assumption: The reservoir (not the quantum computer) is in thermal equilibrium.

From the WCL, A3 is incompatible with A1 (the assumption of fast gates).

“If, on the other hand, we approach the problem from the (physically more consistent) WCL, then A3 is incompatible with A1 (the assumption of fast gates).”

In the WCL the non-Lindbladian terms vanish due to the oscillating character of the e^{i(ω-ω')u} terms in Eq. (19).

“In the WCL the non-Lindbladian terms vanish due to the oscillating character of the e i ( ω -ω ′ ) u terms in Eq. (19).”

In the infinite T limit we recover the case of zero memory-time, i.e., Markovian dynamics.

“In the infinite T limit we then recover the case of zero memory-time, i.e., Markovian dynamics.”

For the periodic case, because differences of Bohr frequencies may be of order 1/Θ, one must average over many periods Θ, requiring t ≫ Θ.

“Timescale analysis : Note that for the periodic case the differences of 'Bohr frequencies' may be of the order of 1 / Θ. Hence we conclude from Eq. (46) that one must average over many periods Θ, i.e., require t ≫ Θ.”

One can relax the time-independence requirement by assuming slowly varying H_S, giving rise to an 'adiabatic MME', Eqs. (35), (36) and (25).

“As we have shown, one can relax this by assuming slowly varying H S , giving rise to an 'adiabatic MME', Eqs. (35), (36) and (25).”

Finite speed of gates implies non-Markovian effects.

“The corollary: finite speed of gates implies non-Markovian effects .”

Strictly speaking the MME (23) is valid only when H_S is not time dependent.

“Strictly speaking the MME (23) is valid only when H S is not time dependent.”

The inequality ||H_C|| ≫ ||H_SR|| is in fact not satisfied in the Markovian WCL, where ||H_SR|| diverges.

“Third, the inequality || H C || ≫|| H SR || is in fact not satisfied in the Markovian WCL, where || H SR || diverges (one should not confuse the small system-reservoir coupling parameter involved in the van-Hove limit with the operator norm, which can be infinite).”

As a consequence of the KMS condition (15) and condition (18), for a generic initial state the system tends to its thermal equilibrium (Gibbs) state at the temperature of the heat bath.

“Namely, as a consequence of the KMS condition (15) and the condition (18), for a generic initial state the system tends to its thermal equilibrium (Gibbs) state at the temperature of the heat bath [35].”

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The SCL rescaling procedure is specifically designed to yield the delta correlation in the limit ε→0.

“The rescaling procedure is specifically designed to yield the delta correlation [Eq. (14)] in the limit as ǫ → 0.”

This adiabatic condition is in contradiction with the fast gate assumption A1.

“This is in contradiction with the fast gate assumption, A1 .”

The singular coupling limit is inconsistent with thermodynamics except in the T→∞ limit.

“Moreover, while mathematically the SCL is rigorous in the scaling limit, it is inconsistent with thermodynamics except in the T → ∞ limit.”

These assumptions are contradictory under the SCL, since the QC would have to be at high T, while the ancillas require low T on the same energy scale E.

“These assumptions are contradictory under the SCL, since the QC would have to be at highT , while the ancillas require lowT on the same energy scale E (set by the Bohr energies of the system = computer + ancillas).”

If ρ_R is at thermal equilibrium at temperature T with respect to H_R, then it must be at thermal equilibrium with respect to H_R/ε^2 at temperature T/ε^2→∞.

“Note that if ρ R is at thermal equilibrium at temperature T with respect to H R , then, since ρ R = exp( -βH R ) /Z is fixed, it must be at thermal equilibrium with respect to H R /ǫ 2 at the temperature T/ǫ 2 →∞ , whence our mention of the high temperature limit, above.”

• AssumptionThe authors state this as an assumption: The reservoir (not the quantum computer) is in thermal equilibrium.

The problem appears for fast gates.

“The problem appears for fast gates.”

Based on computational complexity considerations, a slow realization of gates cannot prevent the inconsistency with the WCL.

“We argue, based on computational complexity considerations, that the answer to this question is negative.”

If we insist that kT ≪ E for the ancillas, then by coupling them to the QC we can no longer assume, in the SCL, that the total system = QC + ancillas is described by Markovian dynamics.

“If we insist that kT ≪ E for the ancillas, then by coupling them to the QC we can no longer assume, in the SCL, that the total system = QC + ancillas is described by Markovian dynamics.”

The Markovian condition F(t) ≃ aδ(t) requires the spectral density to be flat: G(ω)=const.

“For the Markovian condition F ( t ) ≃ aδ ( t ) to hold the spectral density must be flat: G ( ω ) = const.”

A pulse f(t) has a continuous band of frequencies of width Γ≈1/τ_g, with Fourier amplitudes that add to and smear the effective Bohr spectrum {ω}.

“A pulse f ( t ) has a continuous band of frequencies of width Γ ≃ 1 /τ g (where τ g is the gate duration), with amplitudes (Fourier transform) ˆ f (Ω), which add to and smear the effective Bohr spectrum { ω } .”

The SCL is mathematically rigorous in the scaling limit.

“Moreover, while mathematically the SCL is rigorous in the scaling limit, it is inconsistent with thermodynamics except in the T → ∞ limit.”

Merely comparing norms as above does not guarantee adiabaticity.

“Merely comparing norms as above does not guarantee adiabaticity.”

For a fixed Ω_R we do not have a simple Lindblad generator (of the type (2.96) in [40]), but rather a more complicated generator with Lindblad operators depending on the Rabi frequency, as in Eq. (47).

“Hence for a fixed Ω R we do in fact not have a simple Lindblad generator (of the type (2.96) in [40]), but rather a more complicated generator with Lindblad operators depending on the Rabi frequency, as in Eq. (47).”

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The variation of G(ω) happens over the thermal memory time τ_th := 1/kT.

“The variation of G ( ω ) happens over the 'thermal memory' time τ th := 1 /kT .”

Replacing Ω_R by 0 would be correct only if the decay time τ = 1/γ is short enough such that Ω_R τ ≪ 1.

“This would be correct only if the decay time τ = 1 /γ is short enough such that Ω R τ ≪ 1.”

Arbitrarily fast periodic driving (small Θ) is incompatible even with the generalized, finitely localized MME derived in the paper.

“Note that arbitrarily fast periodic driving (small Θ ) is incompatible even with the kind of generalized, finitely localized MME derived here, since then differences of Bohr frequencies matter in Eq. (46)”

For a single frequency Ω, the validity of the Markovian approximation is restricted by condition (46): t ≫ max{1/(ω-ω'+mΩ)}.

“For a single frequency Ω the validity of the Markovian approximation is restricted by the condition (46): t ≫ max { 1 / ( ω -ω ′ + m Ω) } .”

The condition for the adiabatic limit (Markov approximation valid) is that the width of the band is much smaller than the minimal difference of the effective Bohr frequencies.

“Thus, the condition for the adiabatic limit (Markov approximation valid) is: 'the width of the band is much smaller than the minimal difference of the effective Bohr frequencies'.”

The assumptions of fast gates, a constant supply of fresh and cold ancillas, and a Markovian bath may not be mutually consistent.

“We point out that these assumptions may not be mutually consistent in light of rigorous formulations of the Markovian approximation.”

Physically, it is enough to assume that G(ω) is essentially constant over the interval [−ω_0, ω_0] where kT > ω_0 ≫ system's Bohr frequencies.

“Physically, it is enough to assume that G ( ω ) is essentially constant over the interval [ -ω 0 , ω 0 ] where kT &gt; ω 0 ≫ system's Bohr frequencies.”

The Davies approach does not require the high-T limit.

“Hence, while the Davies approach does not require the highT limit, it imposes severe restrictions on the speed of quantum gates.”

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In the case Ω_R τ ≪ 1, the Davies type averaging makes no sense physically.

“However, as explained in Section III B 2, in this case the Davies type averaging makes no sense physically.”

The arbitrary driving field case can be treated qualitatively as a 'superposition' of periodic perturbations.

“It follows from Fourier analysis that this case can be treated qualitatively as a 'superposition' of periodic perturbations discussed above.”

For fast gates and generic environments the dependence of L(t) on the Hamiltonians H_S(u) is non-local, involving memory effects.

“For fast gates and generic environments the dependence is non-local, involving memory effects.”

FT-QEC assumes operations on a time-scale that is short on the scale set by max 1/ω_kl.

“However, FT-QEC assumes operations on a time-scale that is short on the scale set by max ω kl 1 /ω kl .”

For a periodic H_S(t) with a covariant dissipative perturbation, the covariant master equation (35) can be derived.

“One can then derive the covariant master equation (we sketch the derivation below):”

In the WCL only adiabatic Hamiltonian manipulations are allowed.

“Namely, in the WCL only adiabatic Hamiltonian manipulations are allowed.”

The listed alternatives to Markovian FT-QEC are, from the point of view adopted in the paper, free of inconsistencies.

“We have also listed a number of alternatives to Markovian FTQEC which, from the point of view adopted here, are free of inconsistencies.”

The claim that ancillas need not be pure, or could be highly mixed, is not supported by the current standard theory of FT-QEC.

“However, this is not supported by the (current) standard theory of FT-QEC.”

The non-Markovian effects that arise from the application of fast gates in the WCL will in general not be describable by a simple time-local modification such as Λ → Λ U'_i U†_i.

“the non-Markovian effects that arise due to the application of fast gates in the WCL, will in general not be describable by a simple time-local modification such as Λ → Λ U ′ i U † i .”

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The vanishing of an energy scale for degenerate qubits holds, strictly speaking, only for fully adiabatic techniques, e.g., HQC.

“Indeed, the vanishing of an energy scale for degenerate qubits holds, strictly speaking, only for fully adiabatic techniques, e.g., HQC [53, 54].”

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If one works with adiabatic gates to keep the dynamics approximately Markovian, one must slow the gates in proportion to the input size (to some power).

“This means that if one works with adiabatic gates in order to keep the dynamics (approximately) Markovian, the result is that one must slow the gates in proportion to the input size (to some power).”

In the SCL the heating rate is proportional to the square of the coupling strength to the reservoir and hence diverges.

“The heating rate is proportional to the square of the coupling strength to the reservoir, i.e., to 1 /ǫ 2 , and hence diverges in the SCL.”

If one of these errors is a phase-flip, it feeds back via the CNOT gates into the data qubits, producing an error.

“If one of these errors is a phase-flip, it feeds back (via the CNOT gates) into the data qubits, producing an error [65].”

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None of the alternatives to Markovian FT-QEC is comprehensive enough to include the full range of errors one might expect in a full-scale implementation of quantum computing.

“none of these alternatives is so comprehensive as to include the full range of errors one might expect in a full-scale implementation of quantum computing.”

An ancilla initially entangled with the data qubits (violating the assumption of being introduced into the circuit in a tensor-product state) is essentially equivalent to the case of an impure ancilla.

“We note that an ancilla which is initially entangled with the data qubits (violating the assumption of being introduced into the circuit in a tensor-product state) is essentially equivalent to the case of an impure ancilla just described (tracing over the data qubits yields an impure ancilla state).”

• AssumptionRests on an assumption: Ancillas are introduced into the error-correction circuit in a tensor-product state, not entangled with the data qubits.

The fault-tolerant QEC formulation does not allow arbitrarily mixed-state ancillas.

“This formulation, however, does not allow arbitrarily mixed-state ancillas, as argued in the Steane-code example.”

It is not possible to maintain all three assumptions of FT-QEC in either the SCL or the WCL.

“it is not possible to maintain all three assumptions in either the SCL or the WCL.”

In TQC, for low enough temperature, errors on encoded quantum information due to thermal fluctuations are exponentially suppressed.

“This fact implies, for low enough temperature, an exponential suppression of errors on encoded quantum information due to thermal fluctuations.”

TQC is not self-correcting as today's effectively naturally fault-tolerant classical architectures are.

“it is not self-correcting as in today's 'effectively naturally fault-tolerant' classical architectures (see Ref. [84] for an eloquent exposition of this point).”

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Recent results on fault tolerance in non-Markovian settings make a crucial assumption about the smallness of the norm of the system-bath interaction Hamiltonian, which severely restricts the class of physical reservoirs.

“recent results on fault tolerance in non-Markovian settings [19, 20, 21, 22], while representing a significant step forward, make a crucial assumption about the smallness of the norm of the system-bath interaction Hamiltonian, which severely restricts the class of physical reservoirs.”

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In the extension of FT-QEC theory to a non-Markovian setting, the crucial role of fresh and nearly pure ancillas is carried over directly and without change from Markovian FT-QEC theory.

“An important ingredient carried over directly and without change from Markovian FT-QEC theory, is the crucial role of the fresh and nearly pure ancillas [19, 20, 21].”

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Topological features can render the ground state stable against errors represented by local operators that do not involve a number of qubits of the order of the size of the system.

“topological features can render such a ground state stable against errors represented by local operators, namely error operators that do not involve a number of qubits of the order of the size of the system.”

In its present state TQC still requires active intervention in the form of FT-QEC when one tries to compute fault-tolerantly.

“in its present state TQC still requires active intervention, in the form of FT-QEC, when one tries to compute fault-tolerantly.”

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The assumption of a small norm of the system-bath interaction Hamiltonian is not satisfied for some standard models of open systems.

“the important assumption of a small norm of the system-bath interaction Hamiltonian (e.g., Eq. (58) in Ref. [20]) is not satisfied for some standard models of open systems.”

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This violates the threshold condition of FT-QEC, in which the input size and gate times are independent parameters.

“This, however, violates the threshold condition of FT-QEC, in which the input size and gate times are independent parameters (see, e.g., Theorem 12 in [7]).”

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At present there exists an inconsistency in the formulation of the theory of FT-QEC for Markovian baths.

“We therefore conclude that, at present, there exists an inconsistency in the formulation of the theory of FT-QEC for Markovian baths.”

As long as one restricts manipulations to adiabatic ones (thus contradicting A1), quantum computing with degenerate qubits is possible even in the Markovian limit.

“Therefore it appears that as long as one restricts manipulations to adiabatic ones (thus contradicting A1 ), quantum computing with degenerate qubits is possible even in the Markovian limit.”

For degenerate qubit levels, introducing far-detuned couplings with a third auxiliary level yields an effective Hamiltonian H_eff = -(Ω_R^2/Δ)|1><0| + h.c., where Ω_R and Δ are the laser Rabi coupling and detuning.

“For instance, if | 0 〉 and | 1 〉 denote degenerate qubit levels (e.g., hyperfine levels of an ion), one can introduce far-detuned (e.g., laser) couplings of | 0 〉 and | 1 〉 with a third auxiliary level. Second order perturbation theory then yields the effective Hamiltonian H eff = -(Ω 2 R / ∆) | 1 〉〈 0 | +h . c . , where Ω R and ∆ are the laser Rabi coupling and detuning, respectively.”

It does not appear possible at present to relax the assumption of cold ancillas.

“it does not appear possible at present to relax the assumption of cold ancillas.”

FT-QEC is capable of dealing with a gate-modified error model in which faulty gates close to the ideal gates are applied in sequence.

“FT-QEC is capable of dealing with such a (gate-modified) error model.”

The more adiabatic the evolution, the smaller is the probability of the non-Markovian errors per gate.

“Now, it follows from our discussion in Section III B that the more adiabatic the evolution, the smaller is the probability of the non-Markovian errors per gate.”

Assumption A1 (fast gates) is definitely not required in TQC.

“In contrast, A1 is definitely not required in TQC”

Comparing the two expressions for p_non-M, M must grow with input size.

“Comparing the two expressions for p non -M , we see that M must grow with input size.”

Therefore an effective, small but non-vanishing, energy scale E1 := Ω_R^2/Δ is introduced.

“Therefore we see that an effective, small but non-vanishing, energy scale E 1 := Ω 2 R / ∆ is introduced.”

The only two known rigorous general limits leading to Markovian dynamics are the singular coupling limit (SCL), which involves a high temperature limit, and the weak coupling limit (WCL), which requires a constant or adiabatic system Hamiltonian and averaging over long times.

“We have also reviewed the only two known rigorous general limits leading to Markovian dynamics: the singular coupling limit (SCL), which involves taking a high temperature limit, and the weak coupling limit (WCL), which requires either a constant or an adiabatic system Hamiltonian, and averaging over long times in comparison with the inverse of the relevant Bohr frequency.”

The appearance of these new system-energy scales implies that once again the SCL-type contradiction applies.

“At any rate, the appearance of these new system-energy scales implies that once again the SCLtype contradiction applies.”