Guanyu Zhu

It is a major challenge to perform addressable logical operations on constant-rate quantum LDPC (qLDPC) codes. Indeed, the overhead of targeting specific logical qubits represents a crucial bottleneck in many quantum fault-tolerance schemes. We introduce a protocol for performing fully addressable logical $CNOT$ or Hadamard gates with constant quantum space-time overhead, on a family of constant-rate and polynomial-distance qLDPC codes. Specifically, our gadgets perform Hadamard on any chosen logical qubit within a code block, and $CNOT$ between any pair of logical qubits, either within a block or across blocks. We also construct constant-overhead gadgets for highly parallel logical operations, including a large class of permutations of logical qubits. Prior protocols for such operations required polynomial space-time overhead with respect to the distance, or else relied on codes with certain symmetries that lack known asymptotic constructions. Our codes are given by tensor (i.e. hypergraph) products of classical codes constructed from lossless expander graphs. To address individual logical qubits, we develop a constant-overhead code-switching procedure between 2- and 3-dimensional product codes, which generalizes Bombin’s dimensional jump (arXiv:1412.5079). We provide rigorous fault-tolerance proofs for our gadgets, and specifically prove a constant threshold under locally stochastic noise. Along the way, we develop a small-set flip decoder for high-dimensional product codes from lossless expanders. Our techniques yield additional interesting consequences, such as single-shot state preparation of 2-dimensional product codes with constant space-time overhead.

Entanglement renormalization circuits are quantum circuits that can be used to prepare large-scale entangled states. For years, it has remained a mystery whether there exist scale-invariant entanglement renormalization circuits for chiral topological order. In this paper, we solve this problem by demonstrating entanglement renormalization circuits for a wide class of chiral topologically ordered states, including a state sharing the same topological properties as Laughlin's bosonic fractional quantum Hall state at filling fraction 1/4 and eight states with Ising-like non-Abelian fusion rules. The key idea is to build entanglement renormalization circuits by interleaving the conventional multi-scale entanglement renormalization ansatz (MERA) circuit (made of spatially local gates) with quasi-local evolution. Given the miraculous power of this circuit to prepare a wide range of chiral topologically ordered states, we refer to these circuits as MERA with quasi-local evolution (MERAQLE).

Abstract We consider a two-dimensional quantum memory of qubits on a torus encoding an extended Fibonacci string-net model, and construct error correction strategies when those qubits are subjected to depolarizing noise. In the case of a fixed-rate sampling noise model, we find an error correcting threshold of 4.75% with a clustering decoder. Using the concept of tube algebras, we construct a set of measurements and of quantum gates which map arbitrary qubit errors to the Turaev-Viro subspace. Tensor network techniques then allow to quantitatively study the action of Pauli noise on that subspace. We perform Monte-Carlo simulations of the Fibonacci code, and compare the performance of several decoders. To the best of our knowledge, this is the first time that a threshold has been calculated for a two-dimensional error correcting code in which universal quantum computation can be performed in its code space.