Minimal nonorthogonal gate decomposition for qubits with limited control

In quantum control theory, a question of fundamental and practical interest is how an arbitrary unitary transformation can be decomposed into a minimum number of elementary rotations for implementation, subject to various physical constraints. Examples include the singlet-triplet (ST) and exchange-only (EO) qubits in quantum-dot systems, and gate construction in the Solovay–Kitaev algorithm. For two important scenarios, we present complete solutions to the problems of optimal decomposition of single-qubit unitary gates with non-orthogonal rotations. For each unitary gate, the criteria for determining the minimal number of pieces is given, the explicit gate construction procedure, as well as a computer code for practical uses. Our results include an analytic explanation to the four-gate decomposition of EO qubits, previously determined numerically by Divincenzo et al. [Nature (London) 408, 339 (2000)]. Furthermore, compared with the approaches of Ramon sequence and its variant [Phys. Rev. Lett. 118, 216802 (2017)], our method can reduce about 50% of gate time for ST qubits. Finally, our approach can be extended to solve the problem of optimal control of topological qubits, where gate construction is achieved through the braiding operations.

Figure 1

(a) Schematic description of Type-I qubits. Rotation axes are allowed to be chosen freely in a range (orange) bounded by $\widehat{z}$ and $\widehat{m}$ axes. (b) Schematic description of Type-II qubits. Rotation axes are fixed to be either $\widehat{z}$ or $\widehat{m}$. (c) The minimum number of pieces for all possible rotations ($q_{min}$) for Type-I qubits. The color scale represents different $q_{min}$ when $\widehat{m}$ lies in the corresponding area. Panels (d)–(f) show gate time comparisons for ST qubits. Red (upper surface or lines) shows five-piece “Ramon” sequence [19]; blue (middle surface or lines) shows revised schemes proposed by Zhang et al. [21]; green (lower surface or lines) shows minimal decomposition proposed in this work. Target gates are $U(\theta ,\pi /2,\varphi )$, and we set $J_{max}=30h$. Panels (e) and (f) show two-dimensional cut for panel (d) with $\theta =\pi /2$ and $\varphi =\pi$, respectively.

Figure 2

Flowchart for constructing the minimum decomposition sequences for Type-I qubits and Type-II qubits.

Figure 3

Randomized benchmarking for different decomposition schemes for ST qubits. Red (upper) lines show “Ramon” sequence [19]; blue (middle) lines show revised scheme in Ref. [21]; green (lower) lines show minimal decomposition proposed in this work. We set $J_{max}=30$ and $h=1$; ${\sigma }_J/J=0.00426$ (barrier control) for panels (a), (b) and ${\sigma }_J/J=0.0563$ (tilt control) for panels (c), (d); ${\sigma }_h/h=0.575$ for panels (a), (c) and ${\sigma }_h/h=0.288$ for panels (b), (d).