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arXiv:1806.06078 · Physical review. B./Physical review. B

Instantaneous braids and Dehn twists in topologically ordered states

Guanyu Zhu, Ali Lavasani, Maissam Barkeshli

Published Aug 4, 2020• 18 citations
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Abstract

A defining feature of topologically ordered states of matter is the existence of locally indistinguishable states on spaces with nontrivial topology. These degenerate states form a representation of the mapping class group (MCG) of the space, which is generated by braids of defects or anyons, and by Dehn twists along noncontractible cycles. These operations can be viewed as fault-tolerant logical gates in the context of topological quantum error correcting codes and topological quantum computation. Here, we show that braids and Dehn twists can in general be implemented by a constant depth quantum circuit, with a depth that is independent of code distance $d$ and system size. The circuit consists of a constant depth local quantum circuit (LQC) implementing a local geometry deformation of the quantum state, followed by a permutation on (relabeling of) the qubits. The permutation requires permuting qubits that are separated by a distance of order $d$; it can be implemented by collective classical motion of mobile qubits or as a constant depth circuit using long-range SWAP operations (with a range set by $d$) on immobile qubits. We further show that (i) applying a given braid or Dehn twist $k$ times can be achieved with $\mathcal{O}(logk)$ time overhead, independent of code distance and system size, which implies an exponential speedup for certain logical gate sequences by trading space for time, and (ii) an arbitrary element of the MCG can be implemented by a constant depth (independent of $d$) LQC followed by a permutation, where in this case the range of interactions of the LQC grows with the number of generators in the presentation of the group element. Applying these results to certain non-Abelian quantum error correcting codes demonstrates how universal logical gate sets can be implemented on encoded qubits using only constant depth unitary circuits.

DOI
10.1103/physrevb.102.075105

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A defining feature of topologically ordered states of matter is the existence of locally indistinguishable states on spaces with nontrivial topology. These degenerate states form a representation of the mapping class group (MCG) of the space, which is generated by braids of defects or anyons, and by Dehn twists along noncontractible cycles. These operations can be viewed as fault-tolerant logical gates in the context of topological quantum error correcting codes and topological quantum computation. Here, we show that braids and Dehn twists can in general be implemented by a constant depth quantum circuit, with a depth that is independent of code distance ddd and system size. The circuit consists of a constant depth local quantum circuit (LQC) implementing a local geometry deformation of the quantum state, followed by a permutation on (relabeling of) the qubits. The permutation requires permuting qubits that are separated by a distance of order ddd; it can be implemented by collective classical motion of mobile qubits or as a constant depth circuit using long-range SWAP operations (with a range set by ddd) on immobile qubits. We further show that (i) applying a given braid or Dehn twist kkk times can be achieved with O(logk)\mathcal{O}(logk)O(logk) time overhead, independent of code distance and system size, which implies an exponential speedup for certain logical gate sequences by trading space for time, and (ii) an arbitrary element of the MCG can be implemented by a constant depth (independent of ddd) LQC followed by a permutation, where in this case the range of interactions of the LQC grows with the number of generators in the presentation of the group element. Applying these results to certain non-Abelian quantum error correcting codes demonstrates how universal logical gate sets can be implemented on encoded qubits using only constant depth unitary circuits.
A fundamental question in the theory of quantum computation is to understand the ultimate space-time resource costs for performing a universal set of logical quantum gates to arbitrary precision. Here we demonstrate that non-Abelian anyons in Turaev-Viro quantum error correcting codes can be moved over a distance of order of the code distance, and thus braided, by a constant depth local unitary quantum circuit followed by a permutation of qubits. Our gates are protected in the sense that the lengths of error strings do not grow by more than a constant factor. When applied to the Fibonacci code, our results demonstrate that a universal logical gate set can be implemented on encoded qubits through a constant depth unitary quantum circuit, and without increasing the asymptotic scaling of the space overhead. These results also apply directly to braiding of topological defects in surface codes. Our results reformulate the notion of braiding in general as an effectively instantaneous process, rather than as an adiabatic, slow process.
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 space-time overhead known to date.
A fundamental question in the theory of quantum computation is to understand the ultimate space-time resource costs for performing a universal set of logical quantum gates to arbitrary precision. Here we demonstrate that non-Abelian anyons in Turaev-Viro quantum error correcting codes can be moved over a distance of order the code distance, and thus braided, by a constant depth local unitary quantum circuit followed by a permutation of qubits. Our gates are protected in the sense that the length of error strings do not grow by more than a constant factor. When applied to the Fibonacci surface code, our results demonstrate that a universal logical gate set can be implemented on encoded qubits through a constant depth unitary quantum circuit, and without increasing the asymptotic scaling of the space overhead. Our result reformulates the notion of anyon braiding as an effectively instantaneous process, rather than as an adiabatic, slow process.
We present an entirely 2D constant-depth realization of topologically protected phase gates at any level of the Clifford hierarchy, and beyond, using non-Abelian surface codes. Our construction encodes a logical qubit in the quantum double D(G)D(G)D(G) of a non-Abelian group GGG on a triangular spatial patch. The logical gate is implemented by a constant-depth circuit constructed from stacking on the spatial region a symmetry-protected topological (SPT) phase specified by a group 2-cocycle and boundary counter-terms. The Bravyi-König theorem limits the unitary gates implementable by constant-depth quantum circuits on Pauli stabilizer codes in DDD dimensions to the DDD-th level of the Clifford hierarchy. We bypass this limitation, by constructing constant-depth unitary gates at arbitrary levels of the Clifford hierarchy purely in 2D, without sacrificing locality or fault tolerance, at the cost of using the quantum double of a non-Abelian group GGG. Specifically, for G=D4NG = D_{4N}G=D4N​, the dihedral group of order 8N8N8N, we realize the phase gate T1/N=diag(1,eiπ/(4N))T^{1/N} = \mathrm{diag}(1, e^{iπ/(4N)})T1/N=diag(1,eiπ/(4N)) in the logical Z‾\overline{Z}Z basis. In this context, we propose a non-abelian stabilizer group formalism, which we work out for dihedral groups. For 8N=2n8N = 2^n8N=2n, the logical gate lies at the nnn-th level of the Clifford hierarchy and, importantly, has a qubit-only realization: we show that it can be constructed in terms of Clifford-hierarchy stabilizers for a code with nnn physical qubits on each edge of the lattice. We also discuss code-switching to the double surface-code D(Z2×Z2)D(\mathbb{Z}_2\times\mathbb{Z}_2)D(Z2​×Z2​), to complete a universal gate-set in this setup.
Topological quantum computation encodes quantum information in the internal fusion space of non-Abelian anyonic quasiparticles, whose braiding implements logical gates. This goes beyond Abelian topological order (TO) such as the toric code, as its anyons lack internal structure. However, the simplest non-Abelian generalizations of the toric code do not support universality via braiding alone. Here we demonstrate that such minimally non-Abelian TOs can be made universal by treating anyon fusion as a computational primitive. We prepare a 54-qubit TO wavefunction associated with the smallest non-Abelian group, S3S_3S3​, on Quantinuum's H2 quantum processor. This phase of matter exhibits cyclic anyon fusion rules, known to underpin universality, which we evidence by trapping a single non-Abelian anyon on the torus. We encode logical qutrits in the nonlocal fusion space of non-Abelian fluxes and, by combining an entangling braiding operation with anyon charge measurements, realize a universal topological gate set and read-out, which we further demonstrate by topologically preparing a magic state. This work establishes S3S_3S3​ TO as simple enough to be prepared efficiently, yet rich enough to enable universal topological quantum computation.
We study restrictions on locality-preserving unitary logical gates for topological quantum codes in two spatial dimensions. A locality-preserving operation is one which maps local operators to local operators — for example, a constant-depth quantum circuit of geometrically local gates, or evolution for a constant time governed by a geometrically local bounded-strength Hamiltonian. Locality-preserving logical gates of topological codes are intrinsically fault tolerant because spatially localized errors remain localized, and hence sufficiently dilute errors remain correctable. By invoking general properties of two-dimensional topological field theories, we find that the locality-preserving logical gates are severely limited for codes which admit non-abelian anyons, in particular, there are no locality-preserving logical gates on the torus or the sphere with M punctures if the braiding of anyons is computationally universal. Furthermore, for Ising anyons on the M-punctured sphere, locality-preserving gates must be elements of the logical Pauli group. We derive these results by relating logical gates of a topological code to automorphisms of the Verlinde algebra of the corresponding anyon model, and by requiring the logical gates to be compatible with basis changes in the logical Hilbert space arising from local F-moves and the mapping class group.