Towers of generalized divisible quantum codes

Abstract

A divisible binary classical code is one in which every code word has weight divisible by a fixed integer. If the divisor is $2^{\nu }$ for a positive integer $\nu$, then one can construct a Calderbank-Shor-Steane (CSS) code, where $X$-stabilizer space is the divisible classical code, that admits a transversal gate in the $\nu$-th level of Clifford hierarchy. We consider a generalization of the divisibility by allowing a coefficient vector of odd integers with which every code word has zero dot product modulo the divisor. In this generalized sense, we construct a CSS code with divisor $2^{\nu +1}$ and code distance $d$ from any CSS code of code distance $d$ and divisor $2^{\nu }$ where the transversal $X$ is a nontrivial logical operator. The encoding rate of the new code is approximately $d$ times smaller than that of the old code. In particular, for large $d$ and $\nu \ge 2$, our construction yields a CSS code of parameters $[[O(d^{\nu -1}),\Omega (d),d]]$ admitting a transversal gate at the $\nu$-th level of Clifford hierarchy. For our construction we introduce a conversion from magic state distillation protocols based on Clifford measurements to those based on codes with transversal $T$-gates. Our tower contains, as a subclass, generalized triply even CSS codes that have appeared in so-called gauge fixing or code switching methods.

26 pages, 1 figure, (v2) minor changes

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Cited by in corpus (11)

• The cost of universality: A comparative study of the overhead of state distillation and code switching with color codes
• On Optimality of CSS Codes for Transversal $T$
• Classification of Small Triorthogonal Codes
• Designing the Quantum Channels Induced by Diagonal Gates
• Mitigating Coherent Noise by Balancing Weight-2 $Z$-Stabilizers
• Climbing the Diagonal Clifford Hierarchy
• Permutation-Invariant Quantum Codes with Transversal Generalized Phase Gates
• Measurement sequences for magic state distillation
• Transversal Clifford and T-gate codes of short length and high distance
• From Magic State Distillation to Dynamical Systems
• Fault Tolerant Quantum Simulation via Symplectic Transvections