The source of hardware-tailored codes and coding phases

TL;DR

The paper develops a principled route to hardware-tailored quantum codes by identifying an optimal quantum information source that minimizes degradation through a noisy channel, using the Open Random Unitary Model as a testbed. Through variational optimization of the input density matrix, the authors uncover a three-phase source diagram—maximally mixed, $ ext{Z}_2$-symmetric, and no-coding—with first-order transitions and a zero-capacity tricritical point at $(q_U,q_Z)=(0,0.5)$, revealing a deep link between quantum capacity and maximum entropy principles. They construct codes guided by the $ ext{Z}_2$ source, including a classical cat code and a concatenated code equivalent to Shor’s 9-qubit code, and discuss how hardware-aware sources can yield robust thresholds far from the critical point while noting breakdowns near zero capacity. The work connects coding transitions to measurement-induced phase transitions and frames the optimization in a statistical-mechanics language, offering a scalable pathway to design quantum codes for real devices via the corresponding optimal quantum sources and tensor-network–based constructions.

Abstract

A central challenge in quantum error correction is identifying powerful quantum codes tailored to specific hardware and determining their error thresholds above which quantum information is unprotected. This problem is hard because we cannot determine the noise models for our devices. Inspired by the quantum capacity theorem, we seek an optimal quantum source of information, namely the density matrix that degrades minimally when passed through a noisy channel. We explore this idea with the Open Random Unitary Model (ORUM), a simplified model of a $N$-qubit quantum computer with competing depolarizing and dephasing channels as a stand-in for unitary gates and measurements. Through numerical optimization, we find that the ORUM hosts three discrete regimes, three "phases", the "maximally mixed source" phase, a "$\mathbb{Z}_2$ source" phase (where ORUM's $U(1)$ gauge symmetry is broken down to $\mathbb{Z}_2$), and a no-coding phase where all information is lost. These phases exhibit first-order transitions among themselves and converge at a novel zero-capacity multicritical point. These results show a remarkable similarity between the quantum capacity theorem and Jaynes' maximum entropy principle of statistical mechanics. Using the $\mathbb{Z}_2$ source, we build two codes, a classical cat code capable of correcting all the dephasing errors and a concatenated cat code capable of correcting all errors up to a distance $d=\text{min}(m,N)$ and reduces to Shor's 9-qubit code for $m=N=3$. Neither classical nor quantum code survives near the vicinity of the zero-capacity multicritical point in the source phase diagram. Applying our approach to current noisy devices could provide a systematic method for constructing quantum codes for robust computation and communication.

Figures (8)

• Figure 1: Coding transition and superadditivity of the one-qubit depolarizing channel.a, A general depiction of the many-body problem associated with many uses of a quantum channel $\mathcal{N}$. b, A reproduction of the discovery of superadditivity in the coherent information associated with the single qubit depolarizing channel divincenzo_1998_capacity_noisy_channels. While at low noise, the maximally mixed code wins, a 3-qubit cat code yields the highest coherent information $I_c$ at higher $q$, and a 5-qubit cat code wins at still higher $q$. No other cat codes competing out to 11-qubit cat codes. c, The coding phase diagram associated with discontinuities in the quantum capacity out to 10-uses as the depolarizing error rate $q$ is varied.
• Figure 2: Source phase diagram of open random unitary model (ORUM)a, A quantum channel-based master equation for ORUM. b, Single-use/source phase diagram of the ORUM for a single time step showing the maximally mixed code phase, $\mathbb{Z}_2$ source phase, and no-coding regions. The inset shows the width of the $\mathbb{Z}_2$ source phase decays like $1/N$ with the system size $N$ along the dashed black line given by $q_U = 0.2 q_Z$. $\Delta q_U$ and $\Delta q_Z$ are horizontal and vertical components of the line connecting the maximally-mixed source region to the no-coding region along $q_U = 0.2q_Z$. c, Average purity of the optimal state found by the gradient-descent based optimizer showing divergence in optimization time close to the critical point $q_{Zc}=0.5$. d, Dynamical phase diagram of the ORUM obtained by taking $p = 2q_Z/q_U$.
• Figure 3: Classical $\mathbb{Z}_2$ Coding phase diagram.a, Phase diagram of the classical $\mathbb{Z}_2$ code for a 3-qubit random unitary model (ORUM) with periodic quantum error correction (QEC) applied. At long times, the classical $\mathbb{Z}_2$ Code vanishes from the $q_U>0$ region of the phase diagram due to $X$-errors it cannot correct, here depicted by the fading red color. It survives at $q_U=0$ for $q_z \lesssim 0.25$ and $q_z\gtrsim 0.75$ where no $X$-errors arise. The inset shows the computed dynamics of coherent information along $p=2q_Z/q_U$, showing the phase vanishing in constant time along a line cut. b, Coherent information of the classical $\mathbb{Z}_2$ code, scanning along the $q_Z$ axis with $q_U$ fixed at $0$, $0.01$, $0.005$, and $0.01$ respectively for various system sizes $N$ at $t=N$. Finite-size scaling of the $q_U=0$ (top left) plot suggests a second-order phase transition around $q_z=0.25$ (and similarly for $q_z = 0.75$ not shown).
• Figure 4: Perfect communication via many uses of a noisy channela, Output of passing a $N$-qubit code through $m$-uses of a noisy channel $\mathcal{N}$ and a QEC layer after an encoding step $\mathcal{E}$. b, Coherent information $I_c$ of a 3-qubit $\mathbb{Z}_2$ code for $3$-uses of the ORUM channel i.e., $(N,m) = (3,3)$. $I_c$ are shown for $\mathbb{Z}_2$ code with no-QEC (blue), classical $\mathbb{Z}_2$ code with QEC , and the quantum $\mathbb{Z}_2$ code with QEC and repetition $t=9$ (right) for $q_Z$ line cuts along $q_U=0.01$. Quantum $\mathbb{Z}_2$ has a clear advantage over classical $\mathbb{Z}_2$ code. c, Dynamics of the same three cases presented in b at $q_U=0.01, q_Z=0.015$, indicated by the green star, showing improvement in the communication rate at long times by leveraging the bit-flip and phase flip correction. d, A conjectured cartoon phase diagram of the quantum $\mathbb{Z}_2$ code in the thermodynamic limit i.e., $m,N,t \rightarrow \infty$ based on the existence of a QEC threshold for concatenated distance-3 codes aliferis_2005_quantumaccuracy_threshold_concatenated.
• Figure 5: Hardware-tailored code design(a) Variational optimization of a source that maximizes the coherent information, (b) Taking an optimal code for a small system, we can construct more complex codes for practical application by appropriate concatenation of the tableau for stabilizer codes or by using tensor network constructions for other kinds of codes