Quantum Memory and Autonomous Computation in Two Dimensions

TL;DR

The paper demonstrates that autonomous fault-tolerant quantum computation and memory protection are achievable in two spatial dimensions using a fixed, translation-invariant local rule in a quantum cellular automaton framework. By embedding a hierarchical, self-simulating structure atop a measurement-free concatenated quantum code and leveraging Toom-like correction, it proves a nonzero noise threshold below which logical errors vanish with increasing system size, with memory lifetimes diverging in the thermodynamic limit. It also provides a continuous-time Lindbladian realization and shows how arbitrary quantum circuits can be fault-tolerantly encoded and executed with only polylogarithmic overhead. The work advances passive QEC in realistic dimensions and points to practical avenues for self-correcting quantum memories and universal quantum computation without active syndrome extraction or external timing.

Abstract

Standard approaches to quantum error correction (QEC) require active maintenance using measurements and classical processing. The possibility of passive QEC has so far only been established in an unphysical number of spatial dimensions. In this work, we present a simple method for autonomous QEC in two spatial dimensions, formulated as a quantum cellular automaton with a fixed, local and translation-invariant update rule. The construction uses hierarchical, self-simulating control elements based on the classical schemes from the seminal results of Gács (1986, 1989) together with a measurement-free concatenated code. We analyze the system under a local noise model and prove a noise threshold below which the logical errors are suppressed arbitrarily with increasing system size and the memory lifetime diverges in the thermodynamic limit. The scheme admits a continuous-time implementation as a time-independent, translation-invariant local Lindbladian with engineered dissipative jump operators. Further, the recursive nature of our protocol allows for the fault-tolerant encoding of arbitrary quantum circuits and thus constitutes a self-correcting universal quantum computer.

Figures (4)

• Figure 1: Schematic depiction of the noise reduction. Each level of the construction corrects local errors via Toom's rule (structure layer) and a concatenated quantum error-correcting code (data layer) while simultaneously performing a simulation of the quantum cellular automaton acting on the level above. A macro-location is the full space-time region containing operations on one block during the $T$ time-steps required to simulate one update of a single logical cell. The states of simulated higher-level cells are spread out over blocks of qudits on the level below and appear faulty whenever the number of errors within one such block can no longer be corrected by the code. We show that the noise model is effectively rescaled at each level of self-simulation so that the logical noise can be suppressed with increasing system size.
• Figure 2: Self-simulating setup. We construct a fault-tolerant universal quantum cellular automaton $\mathop{\mathrm{\mathsf{R}}}\nolimits$. For a given QCA $\mathop{\mathrm{\mathsf{R}}}\nolimits_2$ which acts on $d$-dimensional qudits, we consider the simulation on $d_\mathrm{Univ}$-dimensional qudits with a known, non-corrected universal QCA $\mathop{\mathrm{\mathsf{Univ}}}\nolimits$. We compile $\mathop{\mathrm{\mathsf{Univ}}}\nolimits$ with a concatenated encoding scheme and implement the resulting schedule in a translation-invariant, time-independent system using locally stored space-time coordinates $(\tau, x,y)$ which are themselves corrected from a local neighborhood. The block-period $M \times M \times T$ that represents one simulated step on the encoded data must be large enough to accommodate the complexity of the simulated automaton $\mathop{\mathrm{\mathsf{R}}}\nolimits_2$. To achieve autonomous error-correction, we analyze the dependencies between the parameters indicated in this Figure and show that there exists a consistent choice with which the system can self-simulate ($\mathop{\mathrm{\mathsf{R}}}\nolimits_2 = \mathop{\mathrm{\mathsf{R}}}\nolimits$).
• Figure 3: Local state-space of the automaton. The quantum cellular automaton $\mathop{\mathrm{\mathsf{R}}}\nolimits$ acts on the local data and structure degrees of freedom which store the quantum information of the universal simulation and the believed space-time coordinates, respectively. In the final definition of the self-correcting automaton, the program information is hard-coded into the transition rules, so it does not need to be stored locally or undergo explicit error-correction.
• Figure 4: Decomposition of the EC-procedure. To prove the structure and data correction properties of our QCA, we decompose each level-$l$ macro-location (red region) into several $M \times M \times T_0$-sized exRecs (orange border) that perform a single layer of encoded $\mathop{\mathrm{\mathsf{Univ}}}\nolimits$-gates. The influence neighborhood (dashed border) of a given exRec includes all surrounding exRecs from within which structural errors can propagate over during one $T_0$-length cycle. The goodness of an exRec is defined based on the structural health and number of faults within its entire influence neighborhood. The number of exRecs within these influence neighborhoods is constant, resulting in a constant power in the noise renormalization, which we absorb using a sufficiently high code distance.

Theorems & Definitions (20)

• Definition 1: Toom's rule
• Lemma 1: proof in the extended version of aharonov1997
• Theorem 1: informal
• Definition 2: Toom's rule for structural information
• Definition 3: Structural health
• Definition 4: Error-correcting simulator
• Definition 5: Local noise
• Lemma 2: Threshold theorem. Proof in aliferis2006, proof-sketch in App. \ref{['app:exRec-review']}
• Definition 6: Sufficient criteria for self-correcting QCA
• Proposition 1: Threshold for self-correcting QCA