We show how to perform scalable fault-tolerant non-Clifford gates in two dimensions by introducing domain walls between the surface code and a non-Abelian topological code whose code space is stabilized by Clifford operators. We formulate a path-integral framework which provides both a macroscopic picture for different logical gates as well as a way to derive the associated microscopic circuits. We also show an equivalence between our approach and prior proposals where a 2D array of qubits reproduces the action of a transversal gate in a 3D stabilizer code over time, thus establishing a connection between 3D codes and 2D non-Abelian topological phases. We prove a threshold theorem for our protocols under local stochastic circuit noise using a just-in-time decoder to correct the non-Abelian code.
We show how to perform scalable fault-tolerant non-Clifford gates in two dimensions by introducing domain walls between the surface code and a non-Abelian topological code whose code space is stabilized by Clifford operators. We formulate a path-integral framework which provides both a macroscopic picture for different logical gates as well as a way to derive the associated microscopic circuits. We also show an equivalence between our approach and prior proposals where a 2D array of qubits reproduces the action of a transversal gate in a 3D stabilizer code over time, thus establishing a connection between 3D codes and 2D non-Abelian topological phases. We prove a threshold theorem for our protocols under local stochastic circuit noise using a just-in-time decoder to correct the non-Abelian code.
We introduce a family of scalable planar fault-tolerant circuits that implement logical non-Clifford operations on a 2D color code, such as a logical T gate or a logical non-Pauli measurement that prepares a magic | T ⟩ state. The circuits are relatively simple, consisting only of physical T gates, C X gates, and few-qubit measurements. They can be implemented with an array of qubits on a 2D chip with nearest-neighbor couplings and no wire crossings. The construction is based on a spacetime path integral representation of a non-Abelian 2+1D topological phase, which is related to the 3D color code. We turn the path integral into a circuit by expressing it as a spacetime Z X tensor network and then traversing it in some chosen time direction. We describe in detail how fault tolerance is achieved using a “just-in-time” decoding strategy, for which we repurpose and extend state-of-the-art color-code matching decoders.
Topological codes have many desirable properties that allow fault-tolerant quantum computation with relatively low overhead. A core challenge for these codes, however, is to achieve a low-overhead universal gate set with limited connectivity. In this work, we explore a non-Pauli stabilizer code that can be used to complete a universal gate set on topological toric and surface codes in strictly two dimensions. Fault-tolerant syndrome extraction for the non-Pauli code requires mid-circuit X corrections, a key difference to conventional Pauli codes. We construct and benchmark a just-in-time (JIT) matching decoder to reliably decide these corrections. Under a phenomenological error model with equally likely physical and measurement errors, we find a high threshold of ≈2.5%, close to the ≈2.9% of a decoder with access to the full syndrome history. We also perform a finite-size scaling analysis to estimate how the logical error rate scales below threshold and verify an exponential suppression in both physical error rate and in the system size. A second global decoding step for Z errors is required and the non-Clifford gates in the circuit reduce the threshold from ≈2.9% to ≈1.8% with a naive decoder. We show how Z decoding can be improved using knowledge of the X corrections, pushing the threshold to ≈2.2%. Our results suggest non-Clifford logic in 2D codes could perform comparably to 2D quantum memory. Our formalism for efficient benchmarking and decoding directly generalizes to a broader family of CSS codes whose X stabilizers are twisted by diagonal Clifford operators, and spacetime versions thereof, defined by CSS-like circuits enriched by CCZ, CS, and T gates.
Quantum computers are far more error-prone than their classical counterparts. Therefore, to build a quantum computer capable of running large-scale quantum algorithms, we must use the techniques of quantum error correction to ensure that the computer produces the correct output even when its components are unreliable. However, the resource requirements of building such a fault-tolerant quantum computer are currently prohibitive. Here, we examine the utility of using three-dimensional (3D) surface codes in a fault-tolerant quantum computer. This family of topological error-correcting codes is a generalization of the well-known 2D surface code to three spatial dimensions. We show that certain 3D surface codes have a transversal logical non-Clifford gate. In a quantum computing architecture, a non-Clifford gate is required to achieve computational universality. Transversal gates do not entangle qubits in different codes, so they are naturally fault tolerant because they do not spread errors. Next, we consider the problem of decoding 3D surface codes. In a quantum error-correcting code, we cannot observe the qubits directly, so we measure parity-check operators to gain information about the state of the code. Decoding is the problem of estimating what error has occurred given a list of unsatisfied parity checks. We observe that 3D surface codes offer asymmetric protection against bit-flip and phase-flip errors, but in both cases, we find that a threshold error rate exists below which we can suppress logical errors by increasing the size of the code. We use our results about logical gates and decoding to propose two fault-tolerant quantum computing architectures that utilize 3D surface codes. Finally, we compare the resource requirements of our architectures with the requirements of leading quantum computing architectures based on topological codes. We find that one of our architectures may be competitive with the leading architectures, depending on the properties of the physical systems used to build the qubits.
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) of a non-Abelian group G 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 D dimensions to the D-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 G. Specifically, for G=D4N, the dihedral group of order 8N, we realize the phase gate T1/N=diag(1,eiπ/(4N)) in the logical Z basis. In this context, we propose a non-abelian stabilizer group formalism, which we work out for dihedral groups. For 8N=2n, the logical gate lies at the n-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 n physical qubits on each edge of the lattice. We also discuss code-switching to the double surface-code D(Z2×Z2), to complete a universal gate-set in this setup.
A fundamental problem in fault-tolerant quantum computation is the tradeoff between universality and dimensionality, exemplified by the the Bravyi-König bound for n-dimensional topological stabilizer codes. In this work, we extend topological Pauli stabilizer codes to a broad class of n-dimensional Clifford hierarchy stabilizer codes. These codes correspond to the (n+1)D Dijkgraaf-Witten gauge theories with non-Abelian topological order. We construct transversal non-Clifford gates through automorphism symmetries represented by cup products. In 2D, we obtain the first transversal non-Clifford logical gates including T and CS for Clifford stabilizer codes, using the automorphism of the twisted Z23 gauge theory (equivalent to D4 topological order). We also combine it with the just-in-time decoder to fault-tolerantly prepare the logical T magic state in O(d) rounds via code switching. In 3D, we construct a transversal logical T gate in a non-Clifford stabilizer code at the third level of the Clifford hierarchy, located on a tetrahedron corresponding to a twisted Z24 gauge theory. Our constructions surpass the Bravyi-König bound by achieving the logical gates in the (n+1)-th level of Clifford hierarchy in n spatial dimension.
Fault-tolerant logic gates will consume a large proportion of the resources of a two-dimensional quantum computing architecture. Here we show how to perform a fault-tolerant non-Clifford gate with the surface code; a quantum error-correcting code now under intensive development. This alleviates the need for distillation or higher-dimensional components to complete a universal gate set. The operation uses both local transversal gates and code deformations over a time that scales with the size of the qubit array. An important component of the gate is a just-in-time decoder. These decoding algorithms allow us to draw upon the advantages of three-dimensional models using only a two-dimensional array of live qubits. Our gate is completed using parity checks of weight no greater than four. We therefore expect it to be amenable with near-future technology. As the gate circumvents the need for magic-state distillation, it may reduce the resource overhead of surface-code quantum computation considerably.