Designing the Quantum Channels Induced by Diagonal Gates

Hu, Jingzhen; Liang, Qingzhong; Calderbank, Robert

doi:10.48550/arxiv.2109.13481

Cited by 2 publications

(11 citation statements)

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“…The Generator Coefficient Framework was introduced in [25] to describe the evolution of stabilizer code states under a physical diagonal gate…”

Section: Generator Coefficient Frameworkmentioning

confidence: 99%

“…The diagonal physical gate U Z preserves a CSS(X, C 2 ; Z, C ⊥ 1 , y) codespace if and only if B µ=0 is a unitary [25], which is equivalent to requiring…”

Section: Generator Coefficient Frameworkmentioning

confidence: 99%

“…We prepare an initial state, apply a physical gate, then measure a code syndrome µ, and finally apply a correction based on µ. For each syndrome, we expand the induced logical operator in the Pauli basis to obtain the generator coefficients [25] that capture state evolution. Intuitively, the diagonal physical gate preserves the code space if and only if the induced logical operator corresponding to the trivial syndrome is unitary.…”

Section: Introductionmentioning

confidence: 99%

“…The action of a diagonal physical operator U Z on code states depends very strongly on the signs of Z-stabilizers [25,27,28] and our generator coefficient framework captures how these signs change the logical operators induced by U Z . For the coherent noise model, a judicious choice of signs creates a decoherence-free subspace, that enables data storage.…”

Section: Introductionmentioning

confidence: 99%

“…In contrast we introduce three basic operations -concatenation, removal of Z-stabilizers, and addition of X-stabilizers -that can be combined to synthesize an arbitrary logical diagonal gate. We present the [[2 m , m r , 2 min{r,m−r} ]] QRM code family [25,29] as a proof of concept.…”

Section: Introductionmentioning

confidence: 99%

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Climbing the Diagonal Clifford Hierarchy

Hu¹,

Liang²,

Calderbank³

2021

Preprint

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Magic state distillation and the Shor factoring algorithm make essential use of logical diagonal gates. We introduce a method of synthesizing CSS codes that realize a target logical diagonal gate at some level l in the Clifford hierarchy. The method combines three basic operations: concatenation, removal of Z-stabilizers, and addition of X-stabilizers. It explicitly tracks the logical gate induced by a diagonal physical gate that preserves a CSS code. The first step is concatenation, where the input is a CSS code and a physical diagonal gate at level l inducing a logical diagonal gate at the same level. The output is a new code for which a physical diagonal gate at level l + 1 induces the original logical gate. The next step is judicious removal of Z-stabilizers to increase the level of the induced logical operator. We identify three ways of climbing the logical Clifford hierarchy from level l to level l + 1, each built on a recursive relation on the Pauli coefficients of the induced logical operators. Removal of Z-stabilizers may reduce distance, and the purpose of the third basic operation, addition of X-stabilizers, is to compensate for such losses. For the coherent noise model, we describe how to switch between computation and storage of intermediate results in a decoherence-free subspace by simply applying Pauli X matrices. The approach to logical gate synthesis taken in prior work focuses on the code states, and results in sufficient conditions for a CSS code to be fixed by a transversal Z-rotation. In contrast, we derive necessary and sufficient conditions by analyzing the action of a transversal diagonal gate on the stabilizer group that determines the code. The power of our approach to logical gate synthesis is demonstrated by two proofs of concept: the [[2 l+1 − 2, 2, 2]] triorthogonal code family, and the [[2 m , m r , 2 min{r,m−r} ]] quantum Reed-Muller code family.

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“…The Generator Coefficient Framework was introduced in [25] to describe the evolution of stabilizer code states under a physical diagonal gate…”

Section: Generator Coefficient Frameworkmentioning

confidence: 99%

“…The diagonal physical gate U Z preserves a CSS(X, C 2 ; Z, C ⊥ 1 , y) codespace if and only if B µ=0 is a unitary [25], which is equivalent to requiring…”

Section: Generator Coefficient Frameworkmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Climbing the Diagonal Clifford Hierarchy

Hu¹,

Liang²,

Calderbank³

2021

Preprint

Self Cite

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Divisible Codes for Quantum Computation

Hu¹,

Liang²,

Calderbank³

2022

Preprint

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Divisible codes are defined by the property that codeword weights share a common divisor greater than one. They are used to design signals for communications and sensing, and this paper explores how they can be used to protect quantum information as it is transformed by logical gates. Given a CSS code C, we derive conditions that are both necessary and sufficient for a transversal diagonal physical operator UZ to preserve C and induce UL. The group of Z-stabilizers in a CSS code C is determined by the dual of a classical [n, k1] binary code C1, and the group of X-stabilizers is determined by a classical [n, k2] binary code C2 that is contained in C1. The requirement that a diagonal physical operator UZ fixes a CSS code C leads to constraints on the congruence of weights in cosets of C2. These constraints are a perfect fit to divisible codes, and represent an opportunity to take advantage of the extensive literature on classical codes with two or three weights. We construct new families of CSS codes using cosets of the first order Reed Muller code defined by quadratic forms. We provide a simple alternative to the standard method of deriving the coset weight distributions (based on Dickson normal form) that may be of independent interest. Finally, we develop an approach to circumventing the Eastin-Knill Theorem which states that no QECC can implement a universal set of logical gates through transversal gates alone. The essential idea is to design stabilizer codes in layers, with N1 inner qubits and N2 outer qubits, and to assemble a universal set of fault tolerant gates on the inner qubits.

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Designing the Quantum Channels Induced by Diagonal Gates

Cited by 2 publications

References 0 publications

Climbing the Diagonal Clifford Hierarchy

Climbing the Diagonal Clifford Hierarchy

Divisible Codes for Quantum Computation

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