We model repetitive quantum error correction (QEC) with the single-error-correcting five-qubit code on a network of individually controlled qubits with always-on Ising couplings. We use our previously designed universal set of quantum gates based on sequences of shaped decoupling pulses. In addition to being accurate quantum gates, the sequences also provide dynamical decoupling (DD) of low-frequency phase noise. The simulation involves integrating the unitary dynamics of six qubits over the duration of tens of thousands of control pulses, using classical stochastic phase noise as a source of decoherence. The combined DD and QEC protocol dramatically improves the coherence, with the QEC alone being responsible for more than an order of magnitude infidelity reduction.
Quantum low-density parity check (qLDPC) codes are among the leading candidates to realize error-corrected quantum memories with low qubit overhead. Potentially high encoding rates and large distance relative to their block size make them appealing for practical suppression of noise in near-term quantum computers. In addition to increased qubit-connectivity requirements compared to more conventional topological quantum error correcting codes, qLDPC codes remain notoriously hard to compute with. In this work, we introduce a construction to implement all Clifford quantum gate operations on the recently introduced lift-connected surface (LCS) codes (Old et al. 2024). These codes can be implemented in a 3D-local architecture and achieve asymptotic scaling
[[n,O(n1/3),O(n1/3)]]. In particular, LCS codes realize favorable instances with small numbers of qubits: For the
[[15,3,3]] LCS code, we provide deterministic fault-tolerant (FT) circuits of the logical gate set
{Hi,Hi,CiXj}i,j∈(0,1,2) based on flag qubits. By adding a procedure for FT magic state preparation, we show quantitatively how to realize an FT universal gate set in
d=3 LCS codes. Numerical simulations indicate that our gate constructions can attain pseudothresholds in the range
pth≈4.8⋅10−3−1.2⋅10−2 for circuit-level noise. The schemes use a moderate number of qubits and are therefore feasible for near-term experiments, facilitating progress for fault-tolerant error corrected logic in high-rate qLPDC codes.
We describe how a universal set of dynamically corrected quantum gates can be implemented using sequences of shaped decoupling pulses on any qubit network forming a sparse bipartite graph with always-on Ising interactions. These interactions are constantly decoupled except when they are needed for two-qubit gates. We analytically study the error operators associated with the constructed gates up to third order in the Magnus expansion, analyze these errors numerically in the unitary time evolution of small qubit clusters, and give a bound on high-order errors for qubits on a large square lattice. We prove that with a large enough toric code the present gate set can be used to implement fault-tolerant quantum memory.
We investigate the possibility of achieving high-fidelity universal two-qubit gates by supplementing optimal tuning of individual qubits with dynamical decoupling (DD) of local
1/f noise. We consider simultaneous local pulse sequences applied during the gate operation and compare the efficiencies of periodic, Carr-Purcell, and Uhrig DD with hard $\ensuremath{\pi}$ pulses along two directions (${\ensuremath{\pi}}_{z/y}$ pulses). We present analytical perturbative results (Magnus expansion) in the quasistatic noise approximation combined with numerical simulations for realistic
1/f noise spectra. The gate efficiency is studied as a function of the gate duration, of the number
n of pulses, and of the high-frequency roll-off. We find that the gate error is nonmonotonic in
n, decreasing as ${n}^{\ensuremath{-}\ensuremath{\alpha}}$ in the asymptotic limit, $\ensuremath{\alpha}\ensuremath{\ge}2$, depending on the DD sequence. In this limit ${\ensuremath{\pi}}_{z}$-Urhig is the most efficient scheme for quasistatic
1/f noise, but it is highly sensitive to the soft UV cutoff. For small number of pulses, ${\ensuremath{\pi}}_{z}$ control yields anti-Zeno behavior, whereas ${\ensuremath{\pi}}_{y}$ pulses minimize the error for a finite
n. For the current noise figures in superconducting qubits, two-qubit gate errors $\ensuremath{\sim}{10}^{\ensuremath{-}6}$, meeting the requirements for fault-tolerant quantum computation, can be achieved. The Carr-Purcell-Meiboom-Gill sequence is the most efficient procedure, stable for
1/f noise with UV cutoff up to gigahertz.
We present a strategy for producing multiqubit gates that promise high fidelity with minimal tuning requirements. Our strategy combines gap protection from the adiabatic theorem with dynamical decoupling in a complementary manner. Energy-level transition errors are protected by adiabaticity and remaining phase errors are mitigated via dynamical decoupling. This is a powerful way to divide and conquer the various error channels. In order to accomplish this without violating a no-go theorem regarding black-box dynamically corrected gates [Phys. Rev. A 80, 032314 (2009)], we require a robust operating point (sweet spot) in control space where the qubits interact with little sensitivity to noise. There are also energy gap requirements for effective adiabaticity. We apply our strategy to an architecture in Si with P donors where we assume we can shuttle electrons between different donors. Electron spins act as mobile ancillary qubits and P nuclear spins act as long-lived data qubits. This system can have a very robust operating point where the electron spin is bound to a donor in the quadratic Stark shift regime. High fidelity single qubit gates may be performed using well-established global magnetic resonance pulse sequences. Single electron-spin preparation and measurement has also been demonstrated. Putting this all together, we present a robust universal gate set for quantum computation.
High-dimensional nuclear spins offer a hardware-efficient route to quantum error correction (QEC), with the spin cat code providing intrinsic robustness against phase errors -- the dominant noise channel in donor-in-silicon architectures. However, realizing the full potential of this encoding requires gate operations that preserve its error-correcting properties. In this work, we construct a universal logical gate set that is error-transparent (ET) to phase errors, and discuss its practical implementations and challenges. The ET gates ensure that phase errors occurring stochastically during gate operations are propagated in a systematically traceable manner and remain correctable in a subsequent QEC step. Among the universal gate set constructed, we identify the logical
X gate as the primary challenge and discuss potential realization schemes. In addition, to fully leverage the spin cat code's advantage over an unencoded qubit, multi-tone microwave driving of the logical
CZ gate is essential. Our simulations show that ET gates significantly outperform non-ET gates and may be necessary to surpass the break-even point. We further show how logical measurement and recovery can be constructed from ET operations, and explain why state-preparation cannot be made ET. In particular, ET measurement in the computational basis is realizable via spin parity measurement, and that error correction circuits constructed from ET operations achieve optimal error correction capacity. Our work charts a concrete path toward full fault-tolerant quantum computation with high-dimensional nuclear spin systems.