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1. Fast and Parallelizable Logical Computation with Homological Product Codes

   https://doi.org/10.1103/PhysRevX.15.021065  

   Xu, Qian; Zhou, Hengyun; Zheng, Guo; Bluvstein, Dolev; Ataides, J_Pablo Bonilla; Lukin, Mikhail D; Jiang, Liang (May 2025, Physical Review X)

   Quantum error correction is necessary to perform large-scale quantum computation but requires extremely large overheads in both space and time. High-rate quantum low-density-parity-check (qLDPC) codes promise a route to reduce qubit numbers, but performing computation while maintaining low space cost has required serialization of operations and extra time costs. In this work, we design fast and parallelizable logical gates for qLDPC codes and demonstrate their utility for key algorithmic subroutines such as the quantum adder. Our gate gadgets utilize transversal logical s between a data qLDPC code and a suitably constructed ancilla code to perform parallel Pauli product measurements (PPMs) on the data logical qubits. For hypergraph product codes, we show that the ancilla can be constructed by simply modifying the base classical codes of the data code, achieving parallel PPMs on a subgrid of the logical qubits with a lower space-time cost than existing schemes for an important class of circuits. Generalizations to 3D and 4D homological product codes further feature fast PPMs in constant depth. While prior work on qLDPC codes has focused on individual logical gates, we initiate the study of fault-tolerant compilation with our expanded set of native qLDPC code operations, constructing algorithmic primitives for preparing $k$-qubit Greenberger-Horne-Zeilinger states and distilling or teleporting $k$magic states with $O\left(1\right)$space overhead in $O\left(1\right)$and $O\left(\sqrt{k}\log k\right)$logical cycles, respectively. We further generalize this to key algorithmic subroutines, demonstrating the efficient implementation of quantum adders using parallel operations. Our constructions are naturally compatible with reconfigurable architectures such as neutral atom arrays, paving the way to large-scale quantum computation with low space and time overheads. Published by the American Physical Society2025 

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2. A family of permutationally invariant quantum codes

   https://doi.org/10.22331/q-2024-04-30-1321  

   Aydin, Arda; Alekseyev, Max A; Barg, Alexander (April 2024, Quantum)

   We construct a new family of permutationally invariant codes that correct $t$Pauli errors for any $t\ge 1$. We also show that codes in the new family correct quantum deletion errors as well as spontaneous decay errors. Our construction contains some of the previously known permutationally invariant quantum codes as particular cases, which also admit transversal gates. In many cases, the codes in the new family are shorter than the best previously known explicit permutationally invariant codes for Pauli errors and deletions. Furthermore, our new code family includes a new $\left(\right(4,2,2\left)\right)$optimal single-deletion-correcting code. As a separate result, we generalize the conditions for permutationally invariant codes to correct $t$Pauli errors from the previously known results for $t=1$to any number of errors. For small $t$, these conditions can be used to construct new examples of codes by computer. 

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3. Quantum circuits for toric code and X-cube fracton model

   https://doi.org/10.22331/q-2024-03-13-1276  

   Chen, Penghua; Yan, Bowen; Cui, Shawn X (March 2024, Quantum)

   We propose a systematic and efficient quantum circuit composed solely of Clifford gates for simulating the ground state of the surface code model. This approach yields the ground state of the toric code in $\lceil 2L+2+lo{g}_2\left(d\right)+\frac{L}{2d}\rceil$time steps, where $L$refers to the system size and $d$represents the maximum distance to constrain the application of the CNOT gates. Our algorithm reformulates the problem into a purely geometric one, facilitating its extension to attain the ground state of certain 3D topological phases, such as the 3D toric model in $3L+8$steps and the X-cube fracton model in $12L+11$steps. Furthermore, we introduce a gluing method involving measurements, enabling our technique to attain the ground state of the 2D toric code on an arbitrary planar lattice and paving the way to more intricate 3D topological phases. 

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4. Adaptive syndrome measurements for Shor-style error correction

   https://doi.org/10.22331/q-2023-08-08-1075  

   Tansuwannont, Theerapat; Pato, Balint; Brown, Kenneth R. (August 2023, Quantum)

   The Shor fault-tolerant error correction (FTEC) scheme uses transversal gates and ancilla qubits prepared in the cat state in syndrome extraction circuits to prevent propagation of errors caused by gate faults. For a stabilizer code of distance $d$that can correct up to $t=\lfloor (d-1)/2\rfloor$errors, the traditional Shor scheme handles ancilla preparation and measurement faults by performing syndrome measurements until the syndromes are repeated $t+1$times in a row; in the worst-case scenario, $(t+1{)}^2$rounds of measurements are required. In this work, we improve the Shor FTEC scheme using an adaptive syndrome measurement technique. The syndrome for error correction is determined based on information from the differences of syndromes obtained from consecutive rounds. Our protocols that satisfy the strong and the weak FTEC conditions require no more than $(t+3{)}^2/4-1$rounds and $(t+3{)}^2/4-2$rounds, respectively, and are applicable to any stabilizer code. Our simulations of FTEC protocols with the adaptive schemes on hexagonal color codes of small distances verify that our protocols preserve the code distance, can increase the pseudothreshold, and can decrease the average number of rounds compared to the traditional Shor scheme. We also find that for the code of distance $d$, our FTEC protocols with the adaptive schemes require no more than $d$rounds on average. 

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5. Deterministic Bethe state preparation

   https://doi.org/10.22331/q-2024-10-24-1510  

   Raveh, David; Nepomechie, Rafael I (October 2024, Quantum)

   We present an explicit quantum circuit that prepares an arbitrary $U\left(1\right)$-eigenstate on a quantum computer, including the exact eigenstates of the spin- $1/2XXZ$quantum spin chain with either open or closed boundary conditions. The algorithm is deterministic, does not require ancillary qubits, and does not require QR decompositions. The circuit prepares such an $L$-qubit state with $M$down-spins using $(\frac{L}{M})-1$multi-controlled rotation gates and $2M(L-M)$CNOT-gates. 

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