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IEEE International Symposium on Information Theory (ISIT)

Co-design of CSS Codes and Diagonal Gates

Jingzhen Hu, Qingzhong Liang, Robert Calderbank

Published Jun 26, 2022• 1 citation
Publisher

Abstract

The challenge of quantum computing is to combine error resilience with universal computation. There are many finite sets of gates that are universal, and a standard choice is to augment the set of Clifford gates by a non-Clifford unitary such as the T gate. Given a CSS code, we introduce a method of synthesizing all possible diagonal physical gates that preserve the codespace and induce a target logical gate. We denote an 〚n, k = k1− k2, d〛 CSS code $\mathcal{C}$ by $\operatorname{CSS} \left( {X,{\mathcal{C}_2};Z,\mathcal{C}_1^ \bot } \right)$, where the [n, k2] binary code ${\mathcal{C}_2}$ determines the X-stabilizers in $\mathcal{C}$, and the [n, n−k1] binary code $\mathcal{C}_1^ \bot $ determines the Z-stabilizers in $\mathcal{C}$. The diagonal entries of a diagonal physical gate are indexed by binary vectors in $\mathbb{F}_2^n$. We show that a diagonal physical gate preserves the CSS codespace if and only if entries from the same coset of ${\mathcal{C}_2}$ in ${\mathcal{C}_1}$ (same X-logical) are identical. We also show that the target logical operator only specifies ${2^{{k_1}}}$ out of 2ndiagonal entries of the diagonal physical gate. The remaining degrees of freedom can be used to optimize implementation of the physical gate within a particular quantum computing infrastructure. This encompasses optimization with respect to locality of the physical gate, a criterion that is essential to fault tolerance. When the target logical operator is the identity, the physical gates that preserve the CSS code represent noise operators to which the codespace is oblivious. We illustrate our method by providing several examples of code-gate pairs for which the target logical gate is a non-Clifford unitary. The framework is extended to stabilizer codes in https://arxiv.org/abs/2109.13481.

DOI
10.1109/isit50566.2022.9834511

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Realizing Logical Diagonal Gates via Transversal Physical ZZZ-Rotations in CSS Codes
Calderbank-Shor-Steane (CSS) codes, constructed from nested classical codes C2⊆C1C_2 \subseteq C_1C2​⊆C1​, are typically optimized for good code parameters. However, practical quantum computing equally demands fault-tolerant logical gates. In this work, we characterize nested pairs (C1,C2)(C_1, C_2)(C1​,C2​) whose resulting CSS codes realize a target logical diagonal gate via transversal physical ZZZ-rotations. In doing so, we recover a result of Camps-Moreno et al. that CSS codes can realize only logical single-qubit ZZZ-rotations and multi-qubit controlled ZZZ-rotations via transversal physical ZZZ-rotations. Building on our characterization, we develop the ''appending construction'', that takes as input an [[n′,k′]][[n',k']][[n′,k′]] CSS code Q′Q'Q′ and a target logical ZZZ-rotation (single-qubit or multi-controlled) ULU_LUL​, and extends Q′Q'Q′ by systematically appending n′′n''n′′ physical qubits to obtain an [[n,k]][[n,k]][[n,k]] CSS code QQQ with n=n′+n′′n = n'+n''n=n′+n′′ and k=k′k=k'k=k′. The target logical gate ULU_LUL​ is realized in QQQ by applying a well-chosen physical transversal ZZZ-rotation to the n′′n''n′′ appended physical qubits. Moreover, any logical gate realized via transversal physical ZZZ-rotations in the input code Q′Q'Q′ remains transversally realizable in the extended code QQQ. The CSS code QQQ may incur a loss in minimum distance, but the loss can be controlled through the parameter choices made in the construction. By repeatedly applying the appending construction, we can extend any CSS code Q′Q'Q′ to obtain a CSS code QQQ that supports fault-tolerant implementations of multiple desired logical ZZZ-rotations. The cost to be paid for this is the increased physical qubit overhead as the number of target logical gates grows. To illustrate our methodology, we construct CSS code families that transversally realize addressable logical single-qubit and multi-controlled ZZZ-rotations.
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Universal fault-tolerant quantum computing combines versatile but expensive operations with specialized but cheap ones. Its efficiency depends on how much computation can be pushed onto the cheap operations and on the size of the code needed to do so. Automorphism gates provide such cheap operations using only physical single-qubit Clifford gates and qubit permutations. Yet no general theory characterizes their maximum logical power or the minimum code size needed to attain it. We develop such a theory. For stabilizer codes encoding k≥3k\geq3k≥3 logical qubits, we show that the largest logical group attainable by automorphisms is generated by all addressable SSS and CX\mathrm{CX}CX gates, and we construct codes attaining it. While this group contains exponentially fewer gates than the full Clifford group, adding one suitable non-Clifford gate yields universality. We further classify the largest logical groups attainable using qubit permutations, physical single-qubit Cliffords, or both across general stabilizer and CSS codes, and derive refined bounds for self-dual CSS subclasses. Achieving the maximum-size logical group through automorphisms requires n=Θ(2k)n=Θ(2^k)n=Θ(2k) physical qubits. By contrast, all addressable diagonal Clifford gates, generated by SSS and CZ\mathrm{CZ}CZ, require only n=Θ(k2)n=Θ(k^2)n=Θ(k2) physical qubits when implemented using physical single-qubit Cliffords alone. Both bounds are tight. This polynomial qubit cost extends beyond Cliffords to all addressable diagonal gates at any fixed level of the Clifford hierarchy, using physical single-qubit diagonal gates. Thus, for full addressability, the sharpest physical-qubit cost divide lies between diagonal and CX\mathrm{CX}CX-type gates, not between Clifford and non-Clifford gates.
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