Fundamental Bounds on Qubit Reset

Qubit reset is a basic prerequisite for operating quantum devices, requiring the export of entropy. The fastest and most accurate way to reset a qubit is obtained by coupling the qubit to an ancilla on demand. Here, we derive fundamental bounds on qubit reset in terms of maximum fidelity and minimum time, assuming control over the qubit and no control over the ancilla. Using the Cartan decomposition of the Lie algebra of qubit plus ancilla, we identify the types of interaction and controls for which the qubit can be purified. For these cases, we show that a time-optimal protocol is resonant purity exchange between qubit and ancilla, where the maximum fidelity is identical for all cases but the minimum time strongly depends on the type of interaction and control.

Qubit reset is a basic prerequisite for operating quantum devices, requiring the export of entropy. The fastest and most accurate way to reset a qubit is obtained by coupling the qubit to an ancilla on demand. Here, we derive fundamental bounds on qubit reset in terms of maximum fidelity and minimum time, assuming control over the qubit and no control over the ancilla. Using the Cartan decomposition of the Lie algebra of qubit plus ancilla, we identify the types of interaction and controls for which the qubit can be purified. For these cases, we show that a time-optimal protocol is resonant purity exchange between qubit and ancilla, where the maximum fidelity is identical for all cases but the minimum time strongly depends on the type of interaction and control.
The ability to initialize qubits from an arbitrary mixed state to a fiducial pure state is a basic building block in quantum information (QI) science [1]. Initializing the qubit, or -equivalently -resetting it after completion of a computational task, requires some means to export entropy. At the same time, for device operation, the qubit needs to be well-protected and isolated from its environment. It is thus not an option to simply let the qubit equilibrate with its environment; rather, active reset is indispensable. A common approach to actively initialize a qubit uses projective measurements [2], but for many QI architectures, this suffers from being slow, see e.g. Refs. [3,4] for the example of superconducting qubits. Rapid reset is made possible by coupling each qubit to an ancilla in a tunable way. This can be a fast decaying state, such as in laser cooling, or an auxiliary system such as another qubit [5,6] or a resonator [7,8]. Then, the coupling strength together with either the switching time for the coupling or the ancilla decay rate determine the overall time required to reset the qubit. This bound on the reset protocol duration is a specific instance of open quantum system speed limits [9]. At present, the overall time required for qubit reset presents a major limitation for device clock speeds in leading QI architectures such as superconducting qubits. On the other hand, in these architectures, there exists a great flexibility in the design of tunable couplings between qubit and ancilla. This raises the question of which type of coupling allows for the most accurate and fastest possible reset.
Here, we answer this question, modelling the ancilla as a two-level system with no further control. Since the ancilla equilibrates with the larger thermal environment only after being decoupled from the qubit, we can consider the reset dynamics to be an element of SU(4) [6,10]. This allows us to leverage earlier results on the quantum optimal control in SU(4) [11][12][13][14]. In particular, we make use of the Cartan decomposition of SU(4) [15,16] to derive a geometric criterion for the time evolution and iden-tify the qubit-ancilla couplings that allow for the purification of the qubit. Then, for all Hamiltonians fulfilling the geometric criterion, we use quantum optimal control theory [17] to determine the controls on the qubit that realize the time-optimal reset. We conjecture, on the basis of numerical simulations, that qubit and ancilla must be resonant for time-optimal reset and analytically derive the minimum time under this condition.
We start by answering the question of what are the conditions on the qubit-ancilla time-evolution operator U such that the qubit (indicated by subscript S for system) can be purified at all? Denoting the ancilla by subscript B (for bath) and employing the Cartan decomposition of SU(4), every element U ∈ SU(4) can be written as [18] with K, K ∈ SU(2) ⊗ SU(2) and σ k the usual Pauli matrices. This representation allows one to separate the evolution operator U into local (K, K ) and non-local (A) parts. In the following, we will refer to the coefficients c k ∈ [0, π] as non-local (NL) coordinates and, for convenience, introduce the notation K = k S ⊗ k B , where k S , k B ∈ SU(2) are the local operations on qubit and ancilla, respectively. Assuming a separable initial state of qubit and ancilla, ρ(0) = ρ S (0) ⊗ ρ B (0), the qubit state at time t is given by where tr B denotes the partial trace over the ancilla, and the dynamical map of the qubit, D ρB(0) , depends parametrically on the initial ancilla state, ρ B (0). A necessary condition for purification of a quantum system is nonunitality of its dynamical map [19] (a dynamical map is called unital if it maps the identity onto itself). In order arXiv:2001.09107v1 [quant-ph] 24 Jan 2020 to check unitality in Eq. (2), we consider the initial state where β ∈ [0, 1] is the ancilla ground state population and γ ∈ C its coherence. Using Eq. (1), we find where is the locally transformed ancilla state. Unitality of D ρB(0) is determined by the partial trace in Eq. (4) since We conclude from Eq. (5) that any U ∈ SU(4), which gives rise to only a single non-vanishing c k necessarily yields a unital map for the qubit, and purification is not possible at all. Occurrence of two non-vanishing NL coordinates is necessary but not yet sufficient for nonunitality of D ρB(0) due to the dependence on ρ B , i.e., on the initial ancilla state ρ B (0), and local operation k B . Non-unitality of D ρB(0) , independent of the ancilla, is guaranteed by three non-vanishing NL coordinates [20].
With this observation, we can relate non-unitality of D ρB(0) with the entangling capability of U for the qubitancilla system. The latter is best analyzed in the Weyl chamber [18] which is a symmetry-reduced version of the cube spanned by c 1 , c 2 , c 3 ∈ [0, π], obtained when eliminating redundancies in Eq. (1). The six symmetries are sketched in the upper part of Fig. 1 with the Weyl chamber shown below. The shaded polyhedron in its center describes all perfectly entangling operations, and the c 1 -axis represents all operations with at most one nonvanishing NL coordinate (which give rise to unital maps for the qubit). The c 1 -axis contains one point of the polyhedron of perfect entanglers-the point L corresponding to the gate cNOT and all gates that are locally equivalent to it, including cPHASE. Albeit being perfect entanglers, cNOT and cPHASE yield unital maps for the qubit. The capability of U to create entanglement between qubit and ancilla is thus a necessary but not a sufficient condition for purification of the qubit.
Next, we determine the qubit-ancilla couplings that allow for purification of the qubit. To this end, we write the qubit-ancilla Hamiltonian,  (6) to the purification condition, stated in terms of the number N c of non-vanishing NL coordinates of the joint qubit-ancilla time evolution, we consider the dynamical Lie algebra L. Its Cartan decomposition, L = k ⊕ p, implies that A in Eq. (1) is given by exp{a} where a is a Cartan subalgebra, i.e., a maximal Abelian subalgebra of p. For the Hamiltonian (6) and the control task of purifying the qubit, it turns out that dim{a} coincides with N c . We can thus determine, already at the level of the algebra and without any further knowledge of the actual dynamics, which combinations ancilla in thermal equilibrium, i.e., without ancilla coherence, In case (ii), we know that time-optimal reset is obtained for a constant field E(t) = E that puts qubit and ancilla into resonance, i.e., E is chosen such that λ 1 − λ 0 = ω B where λ 0 < λ 1 are the eigenvalues of the qubit Hamiltonian H S [6]. A constant resonant field therefore seems to be a suitable pilot for numerical optimization. Figure 2 shows the time-evolution of the qubit purity under a resonant field for one hundred randomly chosen initial qubit states, comparing an analytical approximation (discussed below) and a full numerical solution for cases (i) and (ii). For all initial states, maximum purity occurs at roughly the same point in time for the constant resonant field, cf. Fig. 2(c,f), which we call T min . We now use optimal control theory, specifically Krotov's method [23,24], to find pulse shapes E(t), t ∈ [0, τ ] that maximize P S (τ ) at a given time τ = T min . For a few choices of τ , the red and black dots in Fig. 2(b,e) compare the purities obtained with constant resonant and optimized fields, respectively. In all cases, the optimized fields improve the purity compared to the constant resonant field. For τ < T min , there exists an upper bound for P S (τ ), which is attained (by the constant resonant field) for a coherence-free initial qubit state. In the presence of initial qubit coherence, the field first needs to rotate the coherence into population before swapping with the ancilla [6]. This explains why all optimized fields E(t), similarly to those in Ref. [6], exhibit a strong off-resonant peak in the beginning, followed by the resonant protocol [25]. The dynamical bound for the purity observed in Fig. 2 confirms that T min obtained with the constant resonant field is indeed the minimum time for a complete swap of purities, P S (T min ) = P B (0), and the constant resonant field is a time-optimal choice for qubit purification. When allowing for times τ > T min , a purity swap between qubit and ancilla remains the optimal purification result with the corresponding optimal fields being more complex than the constant resonant solution for τ = T min . Based on these numerical results, we conjecture that time-optimal purification always requires one to choose a constant and resonant field, i.e., E(t) = E such that λ 1 − λ 0 = ω B , independent of O S , O B and O c . We exemplarily demonstrate the procedure to obtain T min analytically for case (i) but emphasize that it works similarly for any other combination of O S , O B , O c ∈ {σ 1 , σ 2 , σ 3 }. In case (i), the eigenvalues of H S are λ 0,1 = ∓ ω 2 S + 4E 2 /2, and the resonant field is given by E = ω 2 B − ω 2 S /2. A closed-form expression for the qubit-ancilla dynamics, itself obtained without any approximation in the SM [22], can be approximated to yield an expression for the time evolution of the qubit purity, where The minimum time for purification, T min = π/(2η), is attained at a local maximum of P S (t), cf. SM [22]. This ex- plains why the evolution of P S (t) for a given qubit purity and ancilla temperature displays a perfect swap of qubit and ancilla purity at T min in Fig. 2(a) independently of the initial qubit state. The impact of the approximation used to derive P S (t) is assessed by comparing Fig. 2 Table I shows exemplary minimal reset times T min for a physical realization of two coupled superconducting qubits [26,27]. It demonstrates that on-demand reset is possible on a time scale of the order of 100 ns -or shorter, depending on the device parameters. The types of coupling and local control give rise to three distinct minimal reset times. The detailed explanation in the SM [22] links the three minimal reset times to a single quantity |A| with T min ≈ π/(2|A|). A can be obtained analyti-   Table I. These re-sults generalize to combinations of Pauli operators, i.e., to O S , O B , O c ∈ span{σ 1 , σ 2 , σ 3 }, cf. SM [22].
To conclude, we have shown that there exists a globally minimal time (among all Hamiltonians) to reset a qubit with maximal fidelity when making use of an ancilla. A time-optimal protocol ensures resonance between qubit and ancilla and swaps their purities. The reset fidelity is thus determined by the initial ancilla purity, making it crucial to engineer a sufficiently high ancilla purity, or, respectively, low ancilla temperature. In experiments with superconducting qubits, for example, this does not pose a fundamental problem [8]. We have also shown that there exists an optimal choice for qubit-ancilla interaction and type of control over the qubit to implement the time-optimal reset. Thanks to the Cartan decomposition of SU(4), this choice can be determined at the level of the algebra, i.e., the Hamiltonian, and does not require knowledge of the actual reset dynamics. Our results thus provide the guiding principles for device design in order to realize the fastest and most accurate protocol for qubit reset in a given QI architecture.
Financial support from the Volkswagenstiftung Project No. 91004 and the ANR-DFG research program COQS (ANR-15-CE30-0023-01, DFG COQS Ko 2301/11-1) is gratefully acknowledged. The work of D.S has been carried out with support of the Technische Universität München Institute for Advanced Study, funded by the German Excellence Initiative.