Addressable gate-based logical computation with quantum LDPC codes

TL;DR

This work introduces a gate-based protocol for addressable single- and multi-qubit Clifford operations on individual logical qubits encoded within one or more quantum LDPC codes and observes that the scheme can be integrated with magic state cultivation protocols to achieve universal, gate-based, and fully addressable quantum computation.

Abstract

Quantum computing relies on quantum error correction for high-fidelity logical operations, but scaling to achieve near-term quantum utility is highly resource-intensive. High-rate quantum LDPC codes can reduce error correction overhead, yet realizing high-rate fault-tolerant computation with these codes remains a central challenge. Apart of the lattice surgery approach, standard schemes for realizing logical gates have so far been restricted to performing global operations on all logical qubits at the same time. Another approach relies on low-rate code switching methods. In this work, we introduce a gate-based protocol for addressable single- and multi-qubit Clifford operations on individual logical qubits encoded within one or more quantum LDPC codes. Our scheme leverages logical transversal operations via an auxiliary Bacon-Shor code to perform logical operations with constant time overhead enabled by teleportation. We demonstrate the implementation of an overcomplete logical Clifford gate set and perform numerical simulations to evaluate the error-correction performance of our protocol. Finally, we observe that our scheme can be integrated with magic state cultivation protocols to achieve universal, gate-based, and fully addressable quantum computation.

Figures (3)

• Figure 1: (a) Example of La-cross code with next-to-nearest neighbor connectivity (labeled as $k=2$). The code is defined by a $n\times n$ lattice with a $(n-k)\times(n-k)$ sublattice with open boundary conditions. A $X$- (blue) and a $Z$-stabilizer (red) are depicted, along with one pair of logical $X_L$ (blue) and $Z_L$ (red) operators. (b) Bacon-Shor code with non-local $X$- (blue) and $Z$-stabilizers (red) spanning along two consecutive rows and columns of the array, respectively. (c) $X$-type (cyan) and $Z$-type (pink) two-body gauge operators of the Bacon-Shor code and its two logical $X_L$ (blue) and $Z_L$ (red) operators. (d) Circuit implementing a logical $\mathcal{R}_y(\pi/2)$ rotation on a single quantum LDPC logical qubit via teleportation and (e) example of implementation via qubit shuttling. Different equivalent representatives of the same logical operator are colored with different gradients of the same color (blue for $X_L$ and red for $Z_L$). (f) Circuit implementing a logical S gate via teleportation. (g) Circuit implementing a logical two-qubit entangling operation between two logical qubits via teleportation.
• Figure 2: (a) Logical operators in a La-cross code can be translated along the main lattice via multiplication by a subset of stabilizers. For $k=2$, an example is illustrated: Dark blue strings are two equivalent representation of the same logical operator $X_L$, while in light blue we show the patter of $X$-stabilizers that translates the logical operator in the first row three rows downward. (b) For $k=2$ codes, for some lattice sizes (i.e., $n=m\cdot n_{min}+1$ for some $m\in\mathbb{N}$) and open boundary conditions, some logical operators can be longer than the code distance $d$. Here we show the case of a $d=2$$k=2$ La-cross code which also admits $d+1=3$-long logical operators. Crucially, it is always possible to find as many non-overlapping equivalent representatives ($X_L$, $X_L'$, $X_L"$ logical strings colored with different gradients of blue) via multiplication by a suitable subset of stabilizers. Notably, some of these equivalent representations can have length $2d$ (e.g, $X_L'$ in the example shown). This observation is key to enable the fault tolerance of our quantum computing gadget.
• Figure 3: Quantum error correction simulations for performing a logical Hadamard (H) rotation on a single logical qubit encoded in a $k=2$ La-cross code. The performance for the Hadamard rotation (blue) is compared against the performance for a memory experiment (orange), showing a moderate increase of the logical error probability and decrease of the physical error threshold. All the results are for a single La-cross logical qubit and for $d$ rounds of error correction. The logical error probability is normalized by the number of rounds and the error bars are standard deviations from the Monte Carlo simulations. The La-cross codes that we have simulated are $[[52,4,4]]$, $[[100,4,5]]$, and $[[202,4,7]]$. BP+OSD decoder was used (see Methods for details).