Protection of quantum systems by nested dynamical decoupling

Abstract

Based on a theorem we establish on dynamical decoupling of time-dependent systems, we present a scheme of nested Uhrig dynamical decoupling (NUDD) to protect multi-qubit systems in generic quantum baths to arbitrary decoupling orders, using only single-qubit operations. The number of control pulses in NUDD increases polynomially with the decoupling order. For general multi-level systems, this scheme can preserve a set of unitary Hermitian system operators which mutually either commute or anti-commute, and hence all operators in the Lie algebra generated from this set of operators, generating an effective symmetry group for the system up to a given order of precision. NUDD can be implemented with pulses of finite amplitude, up to an error in the second order of the pulse durations.

Symmetry-enhanced performance of dynamical decoupling

“…(21). We know from the analytic properties of the QDD sequence that the operators B μ with μ ∈ {x,y,z} all scale at least with τ N min +1 where N min := min(N x ,N z ) [15][16][17], which is supported by numerical results in Ref. [18] and in the present work.…”

“…Thus, the results in Ref. [15] demonstrate the validity of QDD with an evenorder UDD sequence on the inner level. If the inner level has an odd order, the symmetry group generated by MOOS is broken and the scheme based on nested UDD cannot be applied anymore.…”

“…Thus the performance is quantified by the scaling of d with τ , the total duration of the sequence. Recent papers [15][16][17][18] proved the properties of QDD and clarified the dependence of the scaling on the number of pulses N x and N z : The overall scaling of QDD is given by ζ := min{N x ,N z } + 1 independent of the details of the environment (i.e., d ∝ τ ζ ). In this sense QDD is a universal sequence for general decoherence such as UDD is a universal sequence for pure dephasing.…”

“…as shown in Table II [15][16][17]. Hence the analytic bounds on the exponents are sharp for the low-symmetry case.…”

“…Wang et al [15] showed that the effect of QDD can be decomposed in the effects of the inner and the outer sequences. The concept of a mutually orthogonal operation set (MOOS) for nested Uhrig DD was introduced: a set of control operators on the inner level is not affected by a set of control operators on the outer level if both sets come from a MOOS.…”