 Award ID(s):
 1718494
 NSFPAR ID:
 10064523
 Date Published:
 Journal Name:
 IEEE International Symposium on Information Theory
 Format(s):
 Medium: X
 Sponsoring Org:
 National Science Foundation
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Tailored topological stabilizer codes in two dimensions have been shown to exhibit highstoragethreshold error rates and improved subthreshold performance under biased Pauli noise. Threedimensional (3D) topological codes can allow for several advantages including a transversal implementation of nonClifford logical gates, singleshot decoding strategies, and parallelized decoding in the case of fracton codes, as well as construction of fractallattice codes. Motivated by this, we tailor 3D topological codes for enhanced storage performance under biased Pauli noise. We present Clifford deformations of various 3D topological codes, such that they exhibit a threshold error rate of 50% under infinitely biased Pauli noise. Our examples include the 3D surface code on the cubic lattice, the 3D surface code on a checkerboard lattice that lends itself to a subsystem code with a singleshot decoder, and the 3D color code, as well as fracton models such as the Xcube model, the Sierpiński model, and the Haah code. We use the belief propagation with ordered statistics decoder (BP OSD) to study threshold error rates at finite bias. We also present a rotated layout for the 3D surface code, which uses roughly half the number of physical qubits for the same code distance under appropriate boundary conditions. Imposing coprime periodic dimensions on this rotated layout leads to logical operators of weight O(n) at infinite bias and a corresponding exp[−O(n)] subthreshold scaling of the logical failure rate, where n is the number of physical qubits in the code. Even though this scaling is unstable due to the existence of logical representations with O(1) lowrate and O(n2/3) highrate Pauli errors, the number of such representations scales only polynomially for the Clifforddeformed code, leading to an enhanced effective distance.more » « less

A basic question in the theory of faulttolerant 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 TuraevViro 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.more » « less

null (Ed.)Universal quantum computation requires the implementation of a logical nonClifford gate. In this paper, we characterize all stabilizer codes whose code subspaces are preserved under physical T and T † gates. For example, this could enable magic state distillation with nonCSS codes and, thus, provide better parameters than CSSbased protocols. However, among nondegenerate stabilizer codes that support transversal T, we prove that CSS codes are optimal. We also show that triorthogonal codes are, essentially, the only family of CSS codes that realize logical transversal T via physical transversal T. Using our algebraic approach, we reveal new purelyclassical coding problems that are intimately related to the realization of logical operations via transversal T. Decreasing monomial codes are also used to construct a code that realizes logical CCZ. Finally, we use Ax's theorem to characterize the logical operation realized on a family of quantum ReedMuller codes. This result is generalized to finer angle Zrotations in https://arxiv.org/abs/1910.09333.more » « less

Utilizing the framework of
lattice gauge theories in the context of Pauli stabilizer codes, we present methodologies for simulating fermions via qubit systems on a twodimensional square lattice. We investigate the symplectic automorphisms of the Pauli module over the Laurent polynomial ring. This enables us to systematically increase the code distances of stabilizer codes while fixing the rate between encoded logical fermions and physical qubits. We identify a family of stabilizer codes suitable for fermion simulation, achieving code distances of d=2,3,4,5,6,7, allowing correction of any\mathbb{Z}_2 ${\mathbb{Z}}_{2}$ qubit error. In contrast to the traditional code concatenation approach, our method can increase the code distances without decreasing the (fermionic) code rate. In particular, we explicitly show all stabilizers and logical operators for codes with code distances of d=3,4,5. We provide syndromes for all Pauli errors and invent a syndromematching algorithm to compute code distances numerically.\lfloor \frac{d1}{2} \rfloor $\lfloor \frac{d1}{2}\rfloor $ 
The challenge of quantum computing is to combine error resilience with universal computation. Diagonal gates such as the transversal T gate play an important role in implementing a universal set of quantum operations. This paper introduces a framework that describes the process of preparing a code state, applying a diagonal physical gate, measuring a code syndrome, and applying a Pauli correction that may depend on the measured syndrome (the average logical channel induced by an arbitrary diagonal gate). It focuses on CSS codes, and describes the interaction of code states and physical gates in terms of generator coefficients determined by the induced logical operator. The interaction of code states and diagonal gates depends very strongly on the signs of Z stabilizers in the CSS code, and the proposed generator coefficient framework explicitly includes this degree of freedom. The paper derives necessary and sufficient conditions for an arbitrary diagonal gate to preserve the code space of a stabilizer code, and provides an explicit expression of the induced logical operator. When the diagonal gate is a quadratic form diagonal gate (introduced by Rengaswamy et al.), the conditions can be expressed in terms of divisibility of weights in the two classical codes that determine the CSS code. These codes find application in magic state distillation and elsewhere. When all the signs are positive, the paper characterizes all possible CSS codes, invariant under transversal Z rotation through π / 2 l , that are constructed from classical ReedMuller codes by deriving the necessary and sufficient constraints on l . The generator coefficient framework extends to arbitrary stabilizer codes but there is nothing to be gained by considering the more general class of nondegenerate stabilizer codes.more » « less