Markovianity by divisibility
Consider a two-parameter family of quantum channels \({\mathscr{E}}=\{{{\mathcal{E}}}_{{t}_{3},{t}_{1}}| {t}_{3}\ge {t}_{1}\ge 0\}\), where \({{\mathcal{E}}}_{{t}_{3},{t}_{1}}\) describes the time evolution of a quantum system from t1 to t3. This family is called divisible if for any intermediate time t2, the total time evolution can be concatenated, i.e., if for all t3 ≥ t2 ≥ t1 ≥ 0, we have \({{\mathcal{E}}}_{{t}_{3},{t}_{1}}={{\mathcal{E}}}_{{t}_{3},{t}_{2}}\circ {{\mathcal{E}}}_{{t}_{2},{t}_{1}}\)31. We say that the two-parameter family \({\mathscr{E}}\) is Markovian by divisibility if it satisfies the divisibility condition.
A fundamental result holds when the two-parameter family \({\mathscr{E}}\) is differentiable. The Gorini-Kossakowski-Sudarshan-Lindblad (GKSL) theorem31 states that an operator \({{\mathcal{L}}}_{t}\) is the generator of the divisible family of quantum channels if and only if it can be written in the form
$$\frac{{\rm{d}}\rho }{{\rm{d}}t}={{\mathcal{L}}}_{t}[\rho ]=-i[H(t),\rho ]+{{\mathcal{D}}}_{t}[\rho ],$$
(2)
with H(t) Hermitian and with the dissipator
$${{\mathcal{D}}}_{t}[\rho ]=\sum _{a,b}{\Gamma }_{ab}(t)\left[{F}_{a}\rho {F}_{b}^{\dagger }-\frac{1}{2}\{{F}_{b}^{\dagger }{F}_{a},\rho \}\right],$$
(3)
where Γ(t) ≥ 0 (positive semidefinite) for all t ≥ 0.
The division into the Hamiltonian part − i[H(t), ρ] and a dissipative part \({{\mathcal{D}}}_{t}[\rho ]\) is unique if the sums run over a traceless orthogonal operator basis Fa [\({\rm{tr}}({F}_{a}^{\dagger }{F}_{b})\propto {\delta }_{ab}\)]. The Kossakowski matrix Γ can be diagonalized as \({\Gamma }_{ab}(t)={\sum }_{k}{u}_{ak}(t){\gamma }_{k}(t){u}_{bk}^{* }(t)\), yielding the familiar equivalent form for the dissipator \({{\mathcal{D}}}_{t}[\rho ]={\sum }_{k}{\gamma }_{k}(t)\left[{L}_{k}(t)\rho {L}_{k}^{\dagger }(t)-\frac{1}{2}\{{L}_{k}^{\dagger }(t){L}_{k}(t),\rho \}\right],\) with γk(t) ≥ 0, and with jump operators Lk(t) = ∑auak(t)Fa. The operator \({{\mathcal{L}}}_{t}\), at a fixed t, is said to be in Lindblad form if the γk(t) are nonnegative.
The assumption of Markovianity by divisibility is ubiquitous in the modeling of gate-based quantum computing. At the circuit level, the total evolution of the qubits is discretized into time steps, each corresponding to a circuit layer. In general, the state of the system after each time step evolves according to a complex quantum stochastic process35. The assumption of Markovianity by divisibility allows us to approximate this dynamics as an independent sequence of quantum channels. At the gate level, the assumption of Markovianity by divisibility leads to a description in terms of the time-dependent GKSL equation [Eq. (2)]. In this case, H(t) represents the control Hamiltonian that implements the intended unitary operation, while the dissipator captures noise processes such as relaxation and dephasing.
Channel semigroup Markovianity
When the generator appearing in Eq. (2) is time-independent, \({{\mathcal{L}}}_{t}={\mathcal{L}}\), the resulting family of channels forms a one-parameter semigroup, \({\mathscr{E}}=\{{{\mathcal{E}}}_{t}| t\ge 0\}\), with \({{\mathcal{E}}}_{{t}_{2}}\circ {{\mathcal{E}}}_{{t}_{1}}={{\mathcal{E}}}_{{t}_{2}+{t}_{1}}\). In this case, each channel \({{\mathcal{E}}}_{t}\) can be written as \({{\mathcal{E}}}_{t}={e}^{t{\mathcal{L}}}\), with \({\mathcal{L}}\) the generator in Lindblad form. Motivated by this observation, Refs. 32,33 introduced a definition of Markovianity as a property of a single quantum channel, rather than of a family of channels (i.e., of the evolution or process itself). Specifically, a channel \({\mathcal{E}}\) is called Markovian if there exists a generator \({\mathcal{L}}\) in Lindblad form such that \({\mathcal{E}}={e}^{{\mathcal{L}}}\). For clarity, we refer to this notion of Markovianity as channel semigroup Markovianity (CSM). Given an arbitrary channel \({\mathcal{E}}\) one can test whether it is CSM by computing \(\log {\mathcal{E}}\) and verifying that it can be brought to Lindblad form33. Note that the parameterization in Eq. (1) describes a Pauli channel that is CSM by construction.
We note that while it is tempting to relate the concepts of CSM to Markovianity by divisibility, the two describe fundamentally different mathematical objects. From an individual channel characterization perspective, whether it is possible to embed a given quantum channel into a family that is Markovian by divisibility is irrelevant in our case and offers no physical insight into the channel itself.
Pauli twirling
A general quantum channel \({\mathcal{E}}\) admits a Kraus representation of the form \({\mathcal{E}}(\rho )={\sum }_{i}{K}_{i}\rho \,{K}_{i}^{\dagger }\), where the Kraus operators Ki satisfy the completeness relation \({\sum }_{i}{K}_{i}^{\dagger }{K}_{i}={\mathbb{1}}\)36. The n-qubit Pauli words \({P}_{a}\in {\{ {\mathbb{1}},X,Y,Z\}}^{\otimes n}\) form an orthogonal operator basis, satisfying \({\rm{tr}}({P}_{a}{P}_{b})={2}^{n}{\delta }_{ab}\). Each Kraus operator can be expanded in this basis as Ki = ∑aκiaPa, so that we can write any channel as \({\mathcal{E}}(\rho )={\sum }_{ab}{p}_{ab}{P}_{a}\rho {P}_{b}\), with \({p}_{ab}={\sum }_{i}{\kappa }_{ia}{\kappa }_{ib}^{* }\). We say that the matrix pab is the Pauli-basis representation of the channel. In particular, a channel is called a Pauli channel if its Pauli-basis representation is diagonal, that is, if \({\mathcal{E}}(\rho )={\sum }_{a}{p}_{a}{P}_{a}\rho {P}_{a}\), where pa is a probability distribution. Therefore, a Pauli channel is fully characterized by the distribution pa.
An alternative representation of a general channel \({\mathcal{E}}\) is given by its transfer matrix
$${T}_{ab}:=\frac{1}{{2}^{n}}{\rm{tr}}[{P}_{a}{\mathcal{E}}({P}_{b})].$$
(4)
The diagonal elements of T are referred to as the Pauli eigenvalues fa,
$${f}_{a}:=\frac{1}{{2}^{n}}{\rm{tr}}[{P}_{a}{\mathcal{E}}({P}_{a})].$$
(5)
As with the matrix pab, Pauli channels have diagonal transfer matrices, thus, the eigenvalues fa also fully characterize any Pauli channel. The Pauli eigenvalues fa and the probabilities pa are related by the Walsh-Hadamard transform
$${f}_{a}=\sum _{k}{(-1)}^{\langle a,k\rangle }{p}_{k}\rightleftharpoons {p}_{a}=\frac{1}{{4}^{n}}\sum _{k}{(-1)}^{\langle a,k\rangle }{f}_{k},$$
(6)
where the symplectic inner product 〈a, c〉 equals 0 if Pa and Pc commute, and 1 if they anticommute18.
Any quantum channel \({\mathcal{E}}\) can be converted into a Pauli channel \({{\mathcal{E}}}^{{\rm{P}}}\) by Pauli twirling, which is performed by randomly injecting a Pauli word before, and the same random Pauli word after the channel,
$${{\mathcal{E}}}^{{\rm{P}}}(\rho ):=\frac{1}{{4}^{n}}\sum _{a,i}{P}_{a}{K}_{i}{P}_{a}\rho {P}_{a}{K}_{i}^{\dagger }{P}_{a}.$$
(7)
Using again the Pauli basis expansion of the Kraus operators, one finds that \({{\mathcal{E}}}^{{\rm{P}}}\) is indeed a Pauli channel with pa = ∑k∣κka∣2 18. In other words, Pauli twirling removes the off-diagonal elements of the Pauli-basis representation pab. Equivalently, in the transfer-matrix picture, Pauli twirling eliminates the off-diagonal elements of Tab, yielding TP = diag(f).
Generalized Pauli-Lindblad channels
We first present mathematical results, summarized in Fig. 1a, on the relation between Pauli channels and channel semigroup Markovianity. We define PL maps as those maps of the form of Eq. (1) with \({\lambda }_{a}\in {\mathbb{C}}\). It is straightforward to show that there exist PL maps that are not quantum channels. We therefore define the generalized PL channels as those PL maps that are also quantum channels. As we show below, the set of generalized PL channels is essentially equal to the set of Pauli channels.
Fig. 1
a The relations between the maps in this paper. The Pauli-Lindblad (PL) maps (orange, P1) are those maps of the form of Eq. (1) with \({\lambda}_{a}\in {\mathbb{C}}\). The generalized PL channels are defined as those PL maps that are also quantum channels (olive green, P2). This set is essentially equal to the set of Pauli channels (the latter is the closure of the former, with limit points depicted as the boundary, P3). The channel semigroup Markovian channels (teal, CSM) contain channels that are Pauli channels (P4) and non-Pauli channels (P5). The Pauli-Lindblad channels are essentially the CSM channels that are Pauli channels (P4), forming a strict subset of the generalized PL channels. All possible nontrivial effects of Pauli twirling a quantum channel are indicated by the yellow arrows. Pauli twirling can break (CSMB), conserve (CSMC), or instate (CSMI) channel semigroup Markovianity. It can also conserve the non-CSM property of a channel (nCSMC). b The noise channel of a noisy Clifford gate \({\mathcal{U}}\circ {\mathcal{E}}\), with \({\mathcal{U}}(\rho )=U\rho \,{U}^{\dagger }\) the noiseless Clifford, can be Pauli twirled by inserting a random Pauli word Pa before, and the conjugated Pauli word \({P}_{a}^{U}=U{P}_{a}{U}^{\dagger }\) after the noisy gate.
In terms of the PL parameters λa, the Pauli eigenvalues are given by
$${f}_{a}=\exp \left(-2\sum _{k}{\lambda }_{k}\langle a,k\rangle \right),$$
(8)
as was found in Ref. 25. We find that this expression can be inverted,
$${\lambda }_{a}=\frac{1}{{4}^{n}}\sum _{k\ne 0}{(-1)}^{\langle a,k \rangle}\ln ({f}_{k}),$$
(9)
with the definition \({P}_{0}={\mathbb{1}}\). From Eq. (9), it is clear that the PL parameters λa are ill-defined for Pauli channels for which there is an a such that fa = 0. Nevertheless, any such channel can be approached arbitrarily closely by a PL channel. Thus, we have the following result.
Proposition
The set of Pauli channels is the closure of the set of generalized PL channels.
We remark that for the identity channel, the Pauli eigenvalues are f = (1,…,1). As a result, Pauli channels with at least one vanishing Pauli eigenvalue fa = 0 represent strong-noise quantum channels and are not relevant for the characterization of the noise of high-fidelity quantum gates. Moreover, the eigenvalues satisfy fa ∈ [−1, 1]. Therefore, it is even possible for the λa to be complex, which, again, occurs only for strong-noise quantum channels. Consequently, Eq. (9) also shows that the PL parameters λa are not unique for these strong-noise channels, due to the possibility of choosing different branch cuts for the logarithm. This redundancy can be removed by consistently taking the principal branch of the logarithm, thus producing a unique representation in terms of λa for each Pauli channel. Equations (8) and (9) then establish a bijection between the PL parameters λa and Pauli eigenvalues fa (fa ≠ 0).
Given the bijection between the PL parameters and the Pauli eigenvalues, we have the following proposition, which we state due to its importance despite its simplicity. The proposition remains valid for some channels with possible negative Pauli eigenvalues, and in the degenerate case for a single qubit, as long as \({\mathcal{P}}\) has nonzero Pauli eigenvalues; otherwise the PL parameters are ill-defined.
Proposition
A Pauli channel \({\mathcal{P}}\) with strictly positive and nondegenerate Pauli eigenvalues has real and nonnegative PL parameters if and only if \({\mathcal{P}}\) is channel semigroup Markovian (CSM).
Proof
If a Pauli channel \({\mathcal{P}}\) has real and nonnegative PL parameters λ, then \({\mathcal{P}}={e}^{{\mathcal{L}}}\), where \({\mathcal{L}}\) is given in Eq. (1), trivially in Lindblad form. Therefore, \({\mathcal{P}}\) is CSM. For the converse, assume that a Pauli channel \({\mathcal{P}}={e}^{{\mathcal{L}}}\) is CSM with \({\mathcal{L}}\) as in Eq. (1). Suppose that \({\mathcal{P}}\) has a PL parameter λa that is not real and nonnegative. Then it cannot be brought to Lindblad form because λ is unique and \({\mathcal{L}}\) is already in diagonal form with negative or complex eigenvalues λ, which contradicts the assumption that \({\mathcal{P}}\) is CSM. □
The proposition implies that Pauli channels with negative PL parameters (and strictly positive and nondegenerate Pauli eigenvalues) are channel semigroup non-Markovian. This aligns with the concept of non-Markovianity by divisibility in the GKSL equation [Eq. (2)], where it is well-established that a negative rate at any point in time renders the evolution non-Markovian.
Using Eqs. (6) and (9) it is now possible to determine whether an n-qubit Pauli channel characterized by either its probability distribution pa or by its eigenvalues fa is CSM by checking the signs of the parameters λa. For example, for a single-qubit Pauli channel, the criterion according to Eq. (9) is fj ≥ fk fl for all permutations (j, k, l) of (x, y, z). This condition for the single-qubit Pauli channel case was also reported in Ref. 37; our result generalizes this to the arbitrary n-qubit case.
Gate frame
Consider the noisy implementation \(\Phi\) of a unitary quantum gate \({\mathcal{U}}\), with \({\mathcal{U}}(\rho )=U\rho \,{U}^{\dagger }\), where U is the desired unitary. It is always possible to write
$$\Phi ={\mathcal{U}}\circ {\mathcal{E}},$$
(10)
which defines the error channel \({\mathcal{E}}\) of the noisy gate. If U is a Clifford gate, so that \({P}_{a}^{U}U=U{P}_{a}\) for some Pauli word \({P}_{a}^{U}\), the noise channel \({\mathcal{E}}\) can be Pauli-twirled experimentally by picking a Pauli word Pa uniformly at random and inserting the Pauli channel \({{\mathcal{P}}}_{a}(\rho )={P}_{a}\rho {P}_{a}\) before and the channel \({{\mathcal{P}}}_{a}^{U}(\rho )={P}_{a}^{U}\rho {P}_{a}^{U}\) after the noisy gate19,
$$\frac{1}{{4}^{n}}\sum _{a}{{\mathcal{P}}}_{a}^{U}\circ \Phi \circ {{\mathcal{P}}}_{a}={\mathcal{U}}\circ {{\mathcal{E}}}^{{\rm{P}}},$$
(11)
as illustrated in Fig. 1b. We wish to characterize the Pauli-twirled noise \({{\mathcal{E}}}^{{\rm{P}}}\) of a quantum gate as a (generalized) PL channel.
Consider a standard gate-based quantum computer, where a quantum gate, taking time tg, is described by a process that is Markovian by divisibility, i.e., where time evolution is described by Eq. (2). To perform a nontrivial quantum gate, the Hamiltonian H(t) can be time-dependent. To obtain the error channel, we define the gate frame by \({{\tilde{\rho}} (t)}={U}^{\dagger }(t)\rho (t)U(t)\), with ρ(t) the state in the original frame, and dU(t)/dt = − iH(t)U(t). Then we can write
$${\Phi }_{t}={{\mathcal{U}}}_{t}\circ {{\mathcal{E}}}_{t},$$
(12)
with \({{\mathcal{U}}}_{t}(\rho )=U(t)\rho \,{U}^{\dagger }(t)\) the unitary evolution up to time t, and \({{\mathcal{E}}}_{t}\) the error up to time t. The noisy quantum gate, with possible coherent and incoherent errors, is described by the channel \({\Phi }_{{t}_{g}}={{\mathcal{U}}}_{{t}_{g}}\circ {{\mathcal{E}}}_{{t}_{g}}\equiv \Phi\). The error channel \({{\mathcal{E}}}_{t}\) is generated by \({\tilde{{\mathcal{L}}}}_{t}=-i[{\tilde{H}}^{{\prime} }(t),\,\cdot \,]+{\tilde{{\mathcal{D}}}}_{t}\). Here, \({\tilde{H}}^{{\prime} }(t)\) describes any coherent control errors in the gate frame, and \({\tilde{{\mathcal{D}}}}_{t}\) is the gate-frame dissipator, which is obtained by transforming the Kossakowski matrix,
$${\Gamma }_{ab}(t)\to {\tilde{\Gamma }}_{ab}(t)=\sum _{c,d}{T}_{ac}^{U}(t){\Gamma }_{cd}(t){T}_{bd}^{U}(t),$$
(13)
where \({T}_{ab}^{U}(t)\) is the transfer matrix of \({{\mathcal{U}}}_{t}\).
The Kossakowski matrix in the gate frame is thus generally time-dependent, even if it is time-independent in the original frame. Nevertheless, it remains positive semidefinite, so that \({\tilde{{\mathcal{L}}}}_{t}\) generates a family that is Markovian by divisibility. Because of the time dependence, this does not generally imply that \({\mathcal{E}}={{\mathcal{E}}}_{{t}_{g}}\) (which is generated by \(\tilde{{{\mathcal{L}}}_{t}}\)) is CSM, nor does it imply that \({{\mathcal{E}}}^{{\rm{P}}}\) is CSM. In the following, we give examples showing that even if \({\mathcal{E}}\) is CSM, Pauli twirling may break this property (CSMB). We additionally show that Pauli twirling may conserve (CSMC) and instate (CSMI) the CSM property of \({\mathcal{E}}\), and may even conserve non-CSM (nCSMC).
Hadamard dephasing
Consider a qubit channel \({{\mathcal{E}}}_{t}\) generated starting from time t = 0 by a purely dissipative generator [H(t) = 0], with a single jump operator being the Hadamard matrix,
$${L}_{\varphi }=(X+Z)/\sqrt{2},$$
(14)
and associated constant dephasing rate γφ > 0. (In this example, the gate frame equals the original frame since H(t) = 0). Since the jump operator Lφ is time-independent, the channel \({{\mathcal{E}}}_{t}={e}^{t{\mathcal{L}}}={e}^{t{\mathcal{D}}}\) is manifestly CSM for any t ≥ 0. Computing the Pauli eigenvalues fa and using Eq. (9), we find that the PL parameters of \({{\mathcal{E}}}_{t}^{{\rm{P}}}\) are
$$\begin{aligned}{\lambda }_{x}(t)&={\lambda }_{z}(t)=\frac{{\gamma }_{\varphi }t}{2},\\ {\lambda }_{y}(t)&=-\frac{1}{2}\ln [\cosh ({\gamma }_{\varphi }t)].\end{aligned}$$
(15)
We note that λ(t) should be interpreted as specifying the value of the PL parameter for a channel of duration t. For any γφt ≥ 0, the PL parameter λy(t) is negative. Therefore, Pauli twirling may break channel semigroup Markovianity (CSMB). Given a PL qubit channel with two equal PL parameters with value ℓ, the third must be at least \(-\frac{1}{2}\ln [\cosh (2\ell )]\). This is shown by setting two PL parameters to ℓ (e.g. λx = λz = ℓ), and using Eqs. (6) and (8) to find the λy for which pa remains a probability distribution. Hadamard dephasing saturates this bound.
Hadamard dephasing and relaxation
We now extend Hadamard dephasing with a relaxation process. Hadamard dephasing can be viewed as a rotated version of standard dephasing because Lφ = Ry(π/4)ZRy( − π/4), where Ry(ϑ) denotes a rotation along the y-axis by angle ϑ. Analogously, we define a rotated relaxation process with the jump operator L = Ry(π/4)σ−Ry( − π/4), where σ− is the qubit lowering operator, and we denote the corresponding relaxation rate by γ. Considering a purely dissipative generator with jump operators Lφ and L, we obtain the channel \({{\mathcal{E}}}_{t}\). Computing the Pauli eigenvalues and using Eq. (9), we find that the Pauli-twirled channel \({{\mathcal{E}}}_{t}^{{\rm{P}}}\) is parameterized by
$${\lambda }_{x}(t)={\lambda }_{z}(t)=\frac{{\gamma }_{\varphi }t}{2}+\frac{\gamma t}{8},$$
(16)
$${\lambda }_{y}(t)=\frac{\gamma t}{4}-\frac{1}{2}\ln \left[\cosh \left({\gamma }_{\varphi }t-\frac{\gamma t}{4}\right)\right].$$
(17)
This shows that relaxation competes with dephasing in making λy negative, resulting in parameter regimes for which λy ≥ 0 and parameter regimes for which λy < 0. This illustrates that Pauli twirling can also conserve the CSM property of a (non-Pauli) quantum channel (CSMC).
We expand the right-hand side of Eq. (17) around γφ = γ = 0 to second order in γφ and γ. Setting the result to zero, solving for γt, and re-expanding to second order in γφt, we find that the condition for λy(t) < 0 is
$$\gamma t\lesssim {({\gamma }_{\varphi }t)}^{2}.$$
(18)
The relaxation and dephasing rates γ and γφ are related to the relaxation and (Ramsey) dephasing times T1 and \({T}_{2}^{* }\) by γt = t/T1, \({\gamma }_{\varphi }t=t/{T}_{2}^{* }-t/(2{T}_{1})\). Similar steps after first making these substitutions in Eq. (17) lead to
$$\frac{t}{{T}_{1}}\lesssim {\left(\frac{t}{{T}_{2}^{* }}\right)}^{2}.$$
(19)
Interestingly, if the noise is insufficiently biased towards dephasing, this can be compensated by longer idling times.
Although we have thus far introduced Hadamard dephasing and relaxation as a toy model to show negative PL parameters, we stress that it can be realized experimentally under realistic conditions nevertheless. Considering a qubit that undergoes dephasing and relaxation that are Markovian by divisibility, the tilted relaxation and dephasing processes are implemented by applying a π/4 pulse to the qubit, allowing it to idle for a duration t, and then rotating it back with a −π/4 (or 7π/4) pulse. Given any nonzero relaxation rate, one can explore both sides of the parameter regime by sweeping the idling time t and observing the sign change of λy. This gives an exciting opportunity to experimentally observe Pauli-twirling-induced channel semigroup non-Markovianity.
Though experimentally feasible, Hadamard dephasing and relaxation is designed as an illustrative example. Since the unitary \({\mathcal{U}}\) involved is the identity, it does not represent a relevant quantum gate. We therefore proceed by considering a nontrivial quantum gate.
The \(\sqrt{X}\) gate
Consider a qubit subjected to a circularly polarized resonant pulse with drive strength A in the presence of relaxation and dephasing that are Markovian by divisibility. In the frame rotating at the drive frequency, the qubit dynamics is governed by Eq. (2), where the Hamiltonian H = AX/2 is time-independent. The same rotating-frame Hamiltonian is obtained if the pulse is linearly polarized under the rotating wave approximation38. The jump operators consist of Z, with associated time-independent dephasing rate γφ ≥ 0, and the qubit lowering operator \({\sigma }_{-}=\left\vert 0\right\rangle \left.\langle 1\right\vert\), with associated time-independent relaxation rate γ ≥ 0. The closed evolution, i.e., when γ = γφ = 0, describes a rotation along the x-axis, Rx(ϑ), with ϑ = At being the rotation angle. The \(\sqrt{X}\) gate is implemented at tg = π/(2A).
In the gate frame, the time evolution is generated by \({\tilde{{\mathcal{L}}}}_{t}=\tilde{{{\mathcal{D}}}_{t}}\), where the jump operators are \({\tilde{\sigma }}_{-}(t)={e}^{iHt}{\sigma }_{-}{e}^{-iHt}\) and \(\tilde{Z}(t)={e}^{iHt}Z{e}^{-iHt}\), and the decoherence rates are unchanged. The generated family of noise channels is therefore manifestly Markovian by divisibility. However, due to the time-dependent jump operators, \({{\mathcal{E}}}_{{t}_{g}}\) is generally not CSM. Perhaps surprisingly, using the methods from Ref. 33, we find numerically that for certain parameter regions of γφtg, γtg, including 0 ≤ γφtg ≤ 2, 0 ≤ γtg ≤ 2, the noise channel \({{\mathcal{E}}}_{{t}_{g}}\) is CSM nonetheless (see Fig. 2).
Fig. 2: Transition diagram of channel semigroup Markovianity for the error channel of a \(\sqrt{X}\) gate due to Pauli twirling.
The two-dimensional parameter space (γtg, γφtg) is divided into four regions corresponding to the effects of Pauli twirling on channel semigroup Markovianity. In the region labeled as CSMB (red), the initial CSM property is broken by Pauli twirling, and the resulting channel can only be described by a generalized PL channel with a negative PL parameter. In region CSMC (dark blue) CSM is conserved, and the resulting channel has only nonnegative PL parameters. At extreme dephasing rates (γφtg ≳ 2), the error channel is non-CSM; Pauli twirling instates Markovianity in region CSMI (purple) and conserves non-Markovianity in region nCSMC (teal). The solid white line is the boundary separating the regions with different channel semigroup Markovianity after Pauli twirling; the dashed white line depicts the implicit curve λx(tg) = 0 [Eq. (21)], the second-order approximation of this boundary. Most importantly, region CSMB contains experimentally relevant scales, shown in the inset, with excellent agreement between the numerical and analytical second-order boundary.
We calculate the Pauli eigenvalues fa and the corresponding PL parameters λa for the Pauli-twirled error channel \({{\mathcal{E}}}_{{t}_{g}}^{{\rm{P}}}\) utilizing transfer matrix representations and a cumulant expansion (see Methods). To second order [i.e., also ignoring terms proportional to \(\gamma {t}_{g}{({\gamma }_{\varphi }{t}_{g})}^{2}\), \({(\gamma {t}_{g})}^{2}{\gamma }_{\varphi }{t}_{g}\), and \({(\gamma {t}_{g})}^{2}{({\gamma }_{\varphi }{t}_{g})}^{2}\)], we find
$$\begin{aligned}{\lambda }_{x}({t}_{g})&\approx \frac{\gamma {t}_{g}}{4}-\frac{{\sin }^{4}\vartheta }{4{\vartheta }^{2}}{\left({\gamma }_{\varphi }{t}_{g}-\frac{\gamma {t}_{g}}{4}\right)}^{2},\\ {\lambda }_{y,z}({t}_{g})&\approx \frac{\gamma {t}_{g}}{8}\left(1\pm \frac{\sin 2\vartheta }{2\vartheta }\right)+\frac{{\gamma }_{\varphi }{t}_{g}}{2}\left(1\mp \frac{\sin 2\vartheta }{2\vartheta }\right).\end{aligned}$$
(20)
Using the scheme in Fig. 1b to twirl the noise channel of a noisy gate, the above PL parameters can only be observed experimentally at ϑ = mπ/2, because only then the resulting gate is a Clifford gate. For the \(\sqrt{X}\) gate specifically, ϑ = π/2, in which case
$${\lambda }_{x}({t}_{g})\approx \frac{\gamma {t}_{g}}{4}-\frac{1}{{\pi }^{2}}{\left({\gamma }_{\varphi }{t}_{g}-\frac{\gamma {t}_{g}}{4}\right)}^{2},$$
(21)
$${\lambda }_{y}({t}_{g})={\lambda }_{z}({t}_{g})=\frac{\gamma {t}_{g}}{8}+\frac{{\gamma }_{\varphi }{t}_{g}}{2}.$$
(22)
The expressions for λy and λz are exact; at ϑ = π/2 the higher-order terms drop.
Based on Eq. (21), computations as in the previous section result in λx < 0 if
$$\gamma {t}_{g}\lesssim {\left(\frac{2}{\pi }\right)}^{2}{({\gamma }_{\varphi }{t}_{g})}^{2},$$
(23)
or in terms of dephasing and relaxation times,
$$\frac{{t}_{g}}{{T}_{1}}\lesssim {\left(\frac{2}{\pi }\right)}^{2}{\left(\frac{{t}_{g}}{{T}_{2}^{* }}\right)}^{2}.$$
(24)
This is as for Hadamard relaxation and dephasing, but, at equal dephasing times, γtg needs to be smaller by a factor (2/π)2 ≈ 0.4.
Thus, negative PL parameters may be observed for the noise channel of the vanilla implementation of the \(\sqrt{X}\) gate, provided that the noise is sufficiently biased towards dephasing. Platforms whose noise is strongly biased towards dephasing include semiconductor spin-qubit, NV center, ion trap, and neutral atom platforms. As a concrete example, consider spin qubits in silicon, where typically tg ~ 100 ns, T1 ≫ 1 ms, and \({T}_{2}^{* } \sim 1\,\text{-}\,10\,\mu {\rm{s}}\)39. This gives \(\frac{{t}_{g}}{{T}_{1}}\ll 1{0}^{-4}\) and \({\left(\frac{2}{\pi }\right)}^{2}{\left(\frac{{t}_{g}}{{T}_{2}^{* }}\right)}^{2} \sim 1{0}^{-4}\,\text{-}\,1{0}^{-2}\), showing that the condition on the noise bias can be met. If on any given platform the noise bias is insufficient, a possible remedy is pulse stretching, i.e., lowering the drive strength A and thereby increasing tg.