Fundamental bounds on qubit reset

Qubit reset is a key task in the operation of quantum devices which, for many quantum hardware platforms, presently limits device clock speed. While it is known that coupling the qubit to an ancilla on demand allows for the fastest qubit reset, the limits on reset accuracy and speed due to the choice of ancilla have not yet been identified—despite the great flexibility in device design for most quantum hardware platforms. Here, we derive bounds on qubit reset in terms of maximum fidelity and minimum time, assuming control over the qubit and no control over the ancilla. For two-level ancillas, we find a provably time-optimal protocol which consists of purity exchange between qubit and ancilla brought into resonance. The globally minimal time can only be realized for specific choices of coupling and control which we identify. When increasing the size of the ancilla Hilbert space, the maximally achievable fidelity increases, whereas the reset time remains constant. Our results translate into device design principles for realizing, in a given quantum architecture, the fastest and most accurate protocol for qubit reset.

Figure 1

(Color online.) Symmetries (upper part) and construction of the Weyl chamber (lower part) for the characterization of the nonlocal content of any two-qubit operation $\mathsf{U}\in \mathrm{SU}\left(4\right)$, cf. Eqs. (1). The shaded polyhedron within the Weyl chamber describes all perfectly entangling operations. The orange line highlights those $\mathsf{U}$ which lead to unital maps for the qubit. The green line anticipates a time-optimal path for qubit purification identified below. The letters mark specific elements of $\mathrm{SU}\left(4\right)$ referred to in the text.

Figure 2

(Color online.) The time evolution of the qubit purity reveals the minimum time for qubit reset—shown here for the examples ${\mathsf{H}}_{\mathrm{int}}=J({\sigma }_1\otimes {\sigma }_1)$ with ${\mathsf{H}}_{\mathrm{c}}\left(t\right)=\mathcal{E}\left(t\right){\sigma }_1$ (a) and ${\mathsf{H}}_{\mathrm{c}}\left(t\right)=\mathcal{E}\left(t\right){\sigma }_3$ (b). The gray area corresponds to the superimposed evolutions of 50 different initial states $\rho \left(0\right)={\rho }_{\mathrm{S}}\left(0\right)\otimes {\rho }_{\mathrm{B}}\left(0\right)$ with ${\rho }_{\mathrm{B}}\left(0\right)$ the thermal equilibrium state and ${\rho }_{\mathrm{S}}\left(0\right)$ sampled randomly under the condition of identical purity. The red (black) dots indicate the values for ${\mathcal{P}}_{\mathrm{S}}\left(\tau \right)$ obtained with optimized (constant resonant) $\mathcal{E}\left(t\right)$. The dashed blue lines indicate the upper and lower bounds of ${\mathcal{P}}_{\mathrm{S}}$ predicted by Eq. (8). ${\mathcal{P}}_{\mathrm{B}}\left(0\right)$ denotes the initial ancilla purity. The parameters are ${\omega }_{\mathrm{S}}=1$, ${\omega }_{\mathrm{B}}=3$, $J=0.1$, and inverse temperature $\beta =1$.

Figure 3

Purity evolution of a qubit interacting with a qutrit ancilla under a constant, resonant field (blue line). The resonance condition is set with respect to the $|0\rangle \leftrightarrow |1\rangle$ transition in the qutrit, as these are the initially most populated levels. With numerically optimized fields, we also obtained the maximal purity for specific times (red dots). Parameters as in Fig. 2 and ${\omega }_{\mathrm{B},1}=3$ and ${\omega }_{\mathrm{B},2}=2$.

Figure 4

Behavior of $\overline{A}$ and the numerically obtained inverse purification time $1/T_{\min }$ as the control operator is varied via ${\theta }_c$ from a pure ${\sigma }_3$ control at ${\theta }_c=0$ to ${\sigma }_1$-control at ${\theta }_c=\pi /2$ back to a pure ${\sigma }_3$ control at ${\theta }_c=\pi$, but with opposite sign. The other angles are chosen such that there is a ${\sigma }_3\otimes {\sigma }_1$ interaction between qubit and ancilla.

Figure 5

The figure shows the values of $|\overline{A}|$ (left column) and $C$ (right column) for different interactions and control fields. These are determined by ${\theta }_{\mathrm{S}}=\pi /4$ in (a), (b); ${\theta }_{\mathrm{S}}=\pi /2$ in (c), (d); and ${\theta }_{\mathrm{S}}=3\pi /2$ in (e), (f); and the axes represent the angles ${\theta }_{\mathrm{c}}$ and ${\phi }_{\mathrm{c}}-{\phi }_{\mathrm{S}}$. The solid blue line indicates the maximum value of 1 for the respective quantity, $|\overline{A}|$ or $\left|C\right|$, while the dashed line depicts the maximum value for the other quantity ($\left|C\right|$ or $|\overline{A}|$) to ease comparison. Except for (c) and (d), the lines do not coincide and therefore the limit on the energy exchange between qubit and ancilla, which is determined by $C$, cf. Eq. (A22), does not fully explain the physical role of $\overline{A}$.

Figure 6

Sketch of the reshuffling operation on the joint qubit ancilla spectrum that maximizes the qubit purity.