Fault tolerant quantum computation is a critical step in the development of practical quantum computers. Unfortunately, not every quantum error correcting code can be used for fault tolerant computation. Rengaswamy et. al. define CSS-T codes, which are CSS codes that admit the transversal application of the T gate, which is a key step in achieving fault tolerant computation. They then present a family of quantum Reed-Muller fault tolerant codes. Their family of codes admits a transversal T gate, but the asymptotic rate of the family is zero. We build on their work by reframing their CSS-T conditions using the concept of self-orthogonality. Using this framework, we define an alternative family of quantum Reed-Muller fault tolerant codes. Like the quantum Reed-Muller family found by Rengaswamy et. al., our family admits a transversal T gate, but also has a nonvanishing asymptotic rate.\nWe prove three key results in our search for a Reed-Muller CSS-T family with a nonvanishing rate. First, we show an equivalence between a code containing a self-dual subcode and the dual of that code being self-orthogonal. This allows us to more easily determine if a pair of codes define a CSS-T code. Next, we show that if C1 and C2 are both Reed-Muller codes that form a CSS-T code, C1 must be self-orthogonal. This limits the rate of any family that is constructed solely from Reed-Muller codes. Lastly, we define a family of CSS-T codes by choosing C1 = RM(r, 2r + 1) and C2 = RM(0, 2r + 1) for some nonnegative integer r. We show that this family has an asymptotic rate of 1/2, and show that it is the only possible CSS-T family constructed only from Reed-Muller codes where C1 is self dual.
In this work, we study quantum CSS codes with transversal
T gates. Here,
T gate transversality is meant in the strongest sense; the application of physical
T to every physical qubit yields logical
T on every logical qubit, without Clifford corrections. Despite the importance of the
T gate in fault-tolerant quantum computation, the parameters of asymptotic families of such codes have not been improved since the work of Hastings and Haah in 2017, and Haah in 2018. In this work, we significantly broaden the achievable parameters of quantum code families with transversal
T gates, both expanding the regime of achievable polynomial rate and distance, and constructing such codes with constant rate and growing distance; this is the first time the latter has been achieved, even when allowing Clifford corrections after the transversal
T gate. These are also the first codes achieving
γ→0 for a code with a transversal
T gate, where
γ is the overhead exponent of magic state distillation.
To do this, we develop a framework of divisible decreasing monomial codes, punctured at a downward-closed set on the Boolean hypercube to create logical qubits. We prove a closed-form expression for the distance of such a code punctured at such a set, which may be of independent interest. We first instantiate this with an explicit construction based on weighted Reed-Muller codes, puncturing at low Hamming-weight points, and then with a randomised construction, where a small random set of points is protected from the puncturing to save quantum code distance, achieving improved parameters.
This paper introduces a framework for constructing Calderbank-Shor-Steane (CSS) codes that support fault-tolerant logical transversal
Z-rotations. Using this framework, we obtain asymptotically good CSS codes that fault-tolerantly realize the logical transversal Clifford group (i.e., transversal single-qubit Clifford gates are realized within a single code block, while transversal two-qubit Clifford gates are realized across two identical code blocks). Furthermore, investigating CSS-T codes, we: (a) demonstrate asymptotically good CSS-T codes wherein the transversal
T realizes the logical transversal
S†; (b) show that the condition
C2∗C1⊆C1⊥ is necessary but not sufficient for CSS-T codes; and (c) revise the characterizations of CSS-T codes wherein the transversal
T implements the logical identity and the logical transversal
T, respectively.
Abstract CSS-T codes were recently introduced as quantum error-correcting codes that respect a transversal gate. A CSS-T code depends on a CSS-T pair, which is a pair of binary codes
(C1,C2) ( C 1 , C 2 ) such that
C1 C 1 contains
C2 C 2 ,
C2 C 2 is even, and the shortening of the dual of
C1 C 1 with respect to the support of each codeword of
C2 C 2 is self-dual. In this paper, we give new conditions to guarantee that a pair of binary codes
(C1,C2) ( C 1 , C 2 ) is a CSS-T pair. We define the poset of CSS-T pairs and determine the minimal and maximal elements of the poset. We provide a propagation rule for nondegenerate CSS-T codes. We apply some main results to Reed–Muller, cyclic and extended cyclic codes. We characterize CSS-T pairs of cyclic codes in terms of the defining cyclotomic cosets. We find cyclic and extended cyclic codes to obtain quantum codes with better parameters than those in the literature.
We introduce an adaptive simulated annealing algorithm to search for moderately-sized quantum error-correcting codes with high encoding rates and large distances. Our search targets two classes of stabilizer codes: (1) CSS codes, and (2) a subclass of CSS codes that we call ``self-dual with equivalent logicals'' (SWEL) codes, the latter of which which admit transversal implementations of logical Hadamard and phase gates that can be leveraged to construct fault-tolerant gate sets. The search is guided by an energy function that acts as a surrogate for the logical error rate in a code-capacity noise model, combining code distance with a count of minimum-weight logical operators to resolve the discrete plateaux that impede naïve distance optimization. For block lengths of up to
50 physical qubits, our search finds state-of-the-art CSS and SWEL codes whose distances frequently meet or exceed the variants of the quantum Gilbert-Varshamov bound. In addition to providing favorable seed codes for fault-tolerant architectures based on code concatenation, the codes found in this work are promising candidates for high-rate code demonstrations on near-term quantum computing hardware.
Reed-Muller (RM) codes form a classic family studied for its interesting algebraic and combinatorial properties as well as from the perspective of information transmission. They achieve Shannon capacity of the basic binary channel models such as channels with independent erasures or flip errors. They have numerous applications to the theory of computational complexity and have well-understood local testability. They also give rise to a family of quantum codes admitting transversal logical operators in increasing levels of the Clifford hierarchy, a property of codes considered necessary for future fault-tolerant quantum computations. In this dissertation, we show that, despite decades of research surrounding RM codes, new directions sprouting from their simple definition continue to be possible. We will begin by considering an alternate description of RM codes in terms of faces of the Boolean hypercube, and will see that by replacing the hypercube with a more general algebraic object we obtain a large class of previously undiscovered classical error-correcting codes sharing many structural properties with RM codes. These classical codes directly lead to an even larger class of new quantum error-correcting codes with explicitly determined parameters; we will demonstrate that these quantum codes admit many natural transversal logical gates. We finish by specializing to the class of quantum Reed-Muller codes and provide a complete characterization of the logic implemented by these transversal operators, yielding a broad generalization of prior work on the subject.