Dynamically Error-Corrected Gates for Universal Quantum Computation

Scalable quantum computation in realistic devices requires that precise control can be implemented efficiently in the presence of decoherence and operational errors. We propose a general constructive procedure for designing robust unitary gates on an open quantum system without encoding or measurement overhead. Our results allow for a low-level error correction strategy solely based on Hamiltonian engineering using realistic bounded-strength controls and may substantially reduce implementation requirements for fault-tolerant quantum computing architectures.

Figure 1

Left: The Cayley graph for ${\mathbb{Z}}_2\times {\mathbb{Z}}_2$ represented by collective Pauli operators $\{I,X,Y,Z\}$, together with an Eulerian cycle marked by numbers. Edges are labeled by the generators $X$, $Y$, and arrows denote the direction of action. Right: Modified Cayley graph supporting an Eulerian path that cancels errors in a nonidentity gate. The new edges correspond to sequences with the same (leading-order) error.

Figure 2

Quantum circuit for a DCG on qubit 1. The NOOP is implemented by dropping all $C_{\pm 2\theta }$ gates.

Figure 3

DCG performance in a 3-qubit cat-state algorithmic benchmark, using the metric $1-\mathrm{Tr}\sqrt{{\rho }_{\mathrm{cat}}{\rho }_{\mathrm{out}}{\rho }_{\mathrm{cat}}}$, with ${\rho }_{\mathrm{cat}}=|{\psi }_{\mathrm{cat}}⟩⟨{\psi }_{\mathrm{cat}}|$ and ${\rho }_{\mathrm{out}}$ the actual output state. Linear decoherence from a 5-qubit bath and systematic control errors are included. Hadamard and CNOT gates appearing in the inset quantum circuit are decomposed as sequences of (2 and 6, respectively) physical primitive gates, and each such gate is replaced by a DCGs, for a total of $(2+2\times 6)\times 16=256$ control time slots. Every bath spin $I_a$ interacts via a Heisenberg coupling with every other system or bath spin, $H_e=\Gamma \sum _{a<b}{\overrightarrow{I}}^{(a)}·{\overrightarrow{I}}^{(b)}+A\sum _{i,a}{\overrightarrow{\sigma }}^{(i)}·{\overrightarrow{I}}^{(a)}$, in such a way that (in units of ${\tau }_{\min }^{-1}$) $\Gamma =1$ and ${log}_{10}A=-1,-1.4,\dots ,-5.8$, corresponding to different circles along a given curve in the figure. Rectangular pulse profiles are used throughout, systematic pulse-length errors being included by modifying $h_0(t)$ as $h_0(t)(1+ϵ)$. Line thickness proportionally represents error strength $ϵ$. Note the bunching of data points at low EPG for each $ϵ$, signaling a regime dominated by systematic error.