Universal logical gates with constant overhead: instantaneous Dehn twists for hyperbolic quantum codes

Abstract

A basic question in the theory of fault-tolerant quantum computation is to understand the fundamental resource costs for performing a universal logical set of gates on encoded qubits to arbitrary accuracy. Here we consider qubits encoded with constant space overhead (i.e. finite encoding rate) in the limit of arbitrarily large code distance d through the use of topological codes associated to triangulations of hyperbolic surfaces. We introduce explicit protocols to demonstrate how Dehn twists of the hyperbolic surface can be implemented on the code through constant depth unitary circuits, without increasing the space overhead. The circuit for a given Dehn twist consists of a permutation of physical qubits, followed by a constant depth local unitary circuit, where locality here is defined with respect to a hyperbolic metric that defines the code. Applying our results to the hyperbolic Fibonacci Turaev-Viro code implies the possibility of applying universal logical gate sets on encoded qubits through constant depth unitary circuits and with constant space overhead. Our circuits are inherently protected from errors as they map local operators to local operators while changing the size of their support by at most a constant factor; in the presence of noisy syndrome measurements, our results suggest the possibility of universal fault tolerant quantum computation with constant space overhead and time overhead of O(d/ log d). For quantum circuits that allow parallel gate operations, this yields the optimal scaling of spacetime overhead known to date.

Automorphic Bloch theorems for hyperbolic lattices

“…Given these developments, one also anticipates implementations of hyperbolic lattices using other metamaterial platforms such as photonic crystals ( 8 , 9 ). The above concrete realizations of hyperbolic lattices in the laboratory open up vistas for the exploration of quantum mechanics in (negatively) curved space, with possibly far-reaching implications for fundamental physics in the areas of string theory ( 10 – 12 ), quantum gravity ( 13 – 15 ), and quantum information ( 16 – 21 ). In the long-wavelength limit, the Hamiltonian of a quantum particle on a hyperbolic lattice reduces to the well-known Laplace–Beltrami operator on the Poincaré disk ( 22 , 23 ), whose spectrum is well understood.…”

Quantum simulation of hyperbolic space with circuit quantum electrodynamics: From graphs to geometry

“…The setup is highly tunable and can be used to achieve photon interactions, coupling to spin degrees of freedom, or the effects of disorder [37][38][39]. Hyperbolic lattices have been investigated in the context of classical [40][41][42][43][44][45] and quantum spin systems [46], complex networks [47], and recently came into focus due to potential applications for fault-tolerant quantum codes [48][49][50].…”

Automorphic Bloch theorems for hyperbolic lattices

“…Albeit approximate due to experimental limitations, the concrete realization of a hyperbolic lattice in the laboratory opens up new vistas for the exploration of quantum mechanics in (negatively) curved space, with possibly far-reaching implications for fundamental physics in the areas of string theory [4][5][6], quantum gravity [7][8][9], and quantum information [10][11][12][13][14][15]. Besides CQED, one anticipates in the near future the realization of hyperbolic lattices using other platforms such as photonic metama-terials [16,17] and topolectrical circuits [18][19][20], which have been used recently for the design of exotic bandstructures and are well-suited to the implementation of nonstandard lattice geometries.…”