Designing the Quantum Channels Induced by Diagonal Gates

1 Introduction and Review1

• For the Steane code, applying logical Z corrections to every non-trivial syndrome makes the induced logical operator T † for all syndromes, but the distillation protocol no longer converges.The generator coefficients can analyze syndrome probabilities and variants that decode subsets of syndromes.Source: §1 · p. 3
• The framework expresses the logical channel for each syndrome through generator coefficients and separates syndrome probabilities into coefficient-dependent and initial-state-dependent components.The [[4, 2, 2]] example shows both components can depend on Z-stabilizer signs, which can create an embedded decoherence-free subspace.Source: §1 · pp. 3–4
• Changing Z-stabilizer signs in the [[4, 2, 2]] code changes whether syndrome measurement collapses logical information and whether an embedded decoherence-free subspace exists.With positive sign, the subspace is spanned by |01⟩, |10⟩, and |11⟩; with negative sign, logical information does not collapse, but that subspace disappears.Source: §1 · pp. 3–4
• Necessary and sufficient conditions characterize when a diagonal physical gate preserves a CSS code with arbitrary stabilizer signs and describe the resulting logical operator.For transversal Z-rotation through π/2^l, the global condition is equivalent to earlier local trigonometric conditions.Source: §1 · pp. 2, 4
• For quadratic-form diagonal gates, code preservation reduces to divisibility conditions on Hamming weights in the classical codes defining the CSS code.For positive-sign CSS codes and transversal Z-rotation through π/2^l, the paper establishes the necessity of divisibility conditions derived in [26].Source: §1 · p. 4
• The paper characterizes positive-sign CSS codes built from Reed–Muller codes and their punctured or shortened derivatives that are invariant under transversal Z-rotation.It derives necessary and sufficient constraints relating l to the component Reed–Muller code parameters.Source: §1 · p. 4
• For a preserved non-degenerate stabilizer code, the framework constructs a CSS code with the same induced logical operator and Z-distance at least the original distance.The paper concludes that considering non-degenerate stabilizer codes provides no advantage over CSS codes for diagonal gates.Source: §1 · p. 4

2.1 Classical Reed-Muller Codes; 2.2 The MacWilliams Identities

• RM(r, m) consists of evaluations of degree-at-most-r Boolean functions, its dual is RM(m−r−1, m), and its codeword weights have specified powers-of-two divisibility.The highest power of 2 dividing all weights is 2^⌊(m−1)/r⌋; MacWilliams identities relate a code’s weight enumerator to its dual’s.Source: §2.1 · pp. 4–5, §2.2 · p. 5

2.3 The Pauli Group

• The n-qubit Pauli operators are built from single-qubit Paulis, form a matrix basis, and commute or anticommute according to their symplectic inner product.The symplectic inner product determines the commutation sign.Source: §2.3 · p. 5

2.4 The Clifford Hierarchy

• The Clifford hierarchy starts with Paulis, places Clifford operators at level two, and includes the T gate at level three, which can combine with Cliffords for universal computation.The hierarchy is defined recursively by how unitaries conjugate Pauli operators.Source: §2.4 · pp. 5–6

2.5 Stabilizer Codes

• A stabilizer code is the common fixed subspace of a commuting Pauli subgroup; its distance measures the minimum weight outside the stabilizer in its normalizer.For coherent errors such as Z-axis rotations, stabilizer-generator signs can matter, unlike for stochastic independent Pauli errors.Source: §2.5 · p. 6

2.6 CSS Codes

• CSS codes separate X- and Z-type stabilizers and are constructed from nested classical codes C2 ⊂ C1, with code parameters and distance tied to those codes.The CSS code is represented as CSS(X, C2; Z, C⊥1).Source: §2.6 · p. 7

2.7 General Encoding Map for CSS codes; 2.8 General Logical Pauli Operators for CSS codes

• CSS encoding must account for stabilizer signs, while sign choices can also change logical Pauli signs and affect phases in logical-operator combinations.The encoding map incorporates character vectors for the signs of X- and Z-stabilizers.Source: §2.7 · p. 7, §2.8 · p. 8

2.9 Quantum Channels

• A density operator represents an ensemble of pure states, and quantum channels are linear, completely positive, trace-preserving maps with Kraus representations.A channel’s Kraus representation is not unique.Source: §2.9 · p. 9
• Projective measurement gives each outcome a probability and a reduced state, and averaging over outcomes produces an updated density operator.The outcome probability for state |ψx⟩ and projector Πj is ⟨ψx|Πj|ψx⟩.Source: §2.9 · p. 9

3 Generator Coefficients

• Generator coefficients describe diagonal-gate interactions with CSS codes by partitioning binary vectors into stabilizer cosets, and support analysis of average logical channels, code invariance, and induced logical operators.For each syndrome, the induced logical operator is expanded in the Pauli basis; code-space preservation is equivalent to the trivial-syndrome operator being unitary.Source: §3 · pp. 9–10

3.1 Transversal Z-Rotations RZ(θ)

• Transversal Z rotations can realize non-Clifford logical gates when θ is not a multiple of π/2, and the rotation family models coherent noise.The paper uses the Steane-code coefficients as a worked example for the rotation analysis.Source: §3.1 · pp. 10–11
• For transversal RZ(θ), generator coefficients are computed for the Steane code and organized by logical operator and syndrome in Table 1.The seven nontrivial syndromes have equivalent behavior by symmetry; an alternative coefficient definition reduces the calculation to a single coset sum.Source: §3.1 · p. 11, Table 1 · p. 11

3.2 Quadratic Form Diagonal Gates

• Quadratic-form diagonal gates include 1- and 2-local diagonal unitaries in the Clifford hierarchy, with RZ(θ) included for θ = 2π/2^l.Their definition uses a symmetric matrix over integers modulo 2^l.Source: §3.2 · p. 12
• For the all-positive [[4, 2, 2]] code, Table 2 lists generator coefficients for CZ⊗2 at l = 2, while Table 3 lists coefficients for CP⊗2 at l = 3.Source: §3.2 · p. 12, Table 2 · p. 13, Table 3 · p. 13

4 Average Logical Channel

• Generator coefficients describe the average logical channel induced by a diagonal gate on a CSS codespace and extend to arbitrary stabilizer codes.The process begins with a code state, applies the physical gate, measures a syndrome, and applies a syndrome-dependent Pauli correction.Source: §4 · p. 13

4.1 The Kraus Representation

• For the Steane code, syndrome-dependent Pauli Z corrections make the induced logical operator T† for every syndrome under the transversal T gate.The Kraus operators describe the average logical channel; the trivial syndrome induces T†, while nontrivial syndromes require correction.Source: §4.1 · pp. 14–15
• For a quadratic form diagonal gate, the [[4, 2, 2]] code is preserved by CZ⊗2, which induces the logical operator ZL.Source: §4.1 · p. 15

4.2 Probability of Observing Different X-Syndromes

• For the Steane code, syndrome probabilities are independent of the initial code state for every θ and sum to 1.Source: §4.2 · p. 16
• For the Steane code, θ = kπ/2 gives trivial-syndrome probability 1 and preserves the codespace, while θ = π/4 + kπ/2 minimizes that probability.Source: §4.2 · pp. 16–17
• With positive signs, |01⟩, |10⟩, and |11⟩ form an embedded decoherence-free subspace; at θ = π/4 + kπ/2 measurement projects onto it, while changing the Z-stabilizer sign removes it.For the changed sign, syndrome probabilities are independent of the initial state.Source: §4.2 · pp. 17–18

5 CSS codes Preserved by UZ

• Theorem 7 characterizes when a diagonal unitary preserves a CSS codespace; preservation leaves only the zero-syndrome Kraus operator, whose coefficients describe the induced logical operator.For a preserved code, the zero syndrome occurs with probability 1; the induced logical operator is unitary exactly when the preservation condition holds.Source: §5 · p. 18

5.1 QFD Gates

• Theorem 9 specializes preservation to QFD gates through divisibility conditions on quadratic-form values, accommodating arbitrary Z-stabilizer signs via the character vector y.The sign vector shifts the divisibility requirement from a set of codewords to a coset; for R = I, the condition reduces to a Hamming-weight divisibility condition.Source: §5.1 · pp. 18–19

5.2.1 RZ(π/p) and RM Constructions

• For fractional physical Z rotations, CSS-code preservation is characterized by Hamming-weight constraints; with positive stabilizer signs, preservation holds iff 2^(l−1) divides w_H(w∗z) for every w∈C2 and z∈C1.Source: §5.2.1 · pp. 19–20
• For nested Reed–Muller codes C1 = RM(r1,m) ⊃ C2 = RM(r2,m), r1 > r2, Theorem 14 gives an if-and-only-if condition for preserving the positive-stabilizer CSS code under RZ(π/2^(l−1)).The punctured Reed–Muller family includes the [[2^m−1, 1, 3]] triorthogonal codes.Source: §5.2.1 · pp. 20, 22

5.2.2 Generator Coefficients and Trigonometric Identities

• For θ = π/2^l, prior work by Rengaswamy et al. [35] derived stabilizer-code invariance conditions using conjugates of arbitrary Pauli matrices under Z rotations.Source: §5.2.2 · p. 23
• The generator-coefficient constraint is equivalent to the first trigonometric condition of Rengaswamy [35, Theorem 17], which implies the second and generalizes the first to arbitrary θ.Unlike the local trigonometric conditions, the square-sum constraint on generator coefficients is global.Source: §5.2.2 · pp. 23–24
• The equivalence is established using the MacWilliams identities, and the analysis extends from CSS codes to general stabilizer codes.Source: §5.2.2 · p. 23
• Theorem 20 gives a necessary and sufficient generator-coefficient condition for RZ(θ) to realize a logical operation on a CSS codespace.The condition is stated for all nonzero w in C2.Source: §5.2.2 · p. 23

6 Conclusion

• Generator coefficients describe code-state evolution through a diagonal gate and syndrome-dependent correction, while accounting for the signs of CSS-code Z-stabilizers.Source: §6 · p. 24
• The paper derives necessary and sufficient conditions for diagonal gates to preserve CSS codes and gives an explicit expression for the induced logical operator.Source: §6 · p. 24
• For transversal Z-rotations, preservation can be expressed through weight divisibility in the two classical codes defining a CSS code.Source: §6 · p. 24
• With positive signs, the paper establishes necessary and sufficient conditions for Reed–Muller component codes to form CSS families invariant under transversal Z-rotations.Source: §6 · p. 24
• The framework applies to stabilizer codes with arbitrary signs, while constraints for CSS codes from decreasing monomial codes remain open.Source: §6 · p. 24

Case 1

• The generator coefficient framework computes output error by tracking operators, whereas Reichardt calculated it by tracking code-state evolution.Source: A Magic State Distillation Using the… · p. 27

Noise model

• For noisy inputs, the analysis assumes equal-probability dephasing on each of the seven physical qubits and treats success as including undetectable errors or errors that cancel the observed syndrome.Source: A Magic State Distillation Using the… · p. 27

Case 1

• The Steane-code initial-error threshold is 0.1464..., compared with 0.1415.. for the [[15, 1, 3]] code.Source: A Magic State Distillation Using the… · p. 28

Case 2

• Case 2 uses syndrome-dependent corrections to obtain logical T † with probability 1 when the physical transversal T is exact.Source: A Magic State Distillation Using the… · p. 28
• The Case 2 protocol does not converge because its output error rate does not fall below y = x in the positive orthant.Source: A Magic State Distillation Using the… · p. 28

Case 3

• Correcting only one non-trivial syndrome in Case 3 raises success probability slightly, but the output-error linear-term prefactor exceeds 1, so the protocol does not converge.Source: A Magic State Distillation Using the… · p. 28

Scope

• The analysis is also applicable to codes perfectly preserved by transversal T, including the [[15, 1, 3]] code, and informs the success-probability–fidelity trade-off.Source: A Magic State Distillation Using the… · p. 28

B Generator Coefficient Framework for Stabilizer codes

• For general stabilizer codes, the generalized generator coefficients describe the same average logical channel, with logical Pauli Z representatives chosen from T⊥/J up to sign.Source: B Generator Coefficient Framework for… · p. 29
• The construction yields a CSS code with n′ = n, k′ = k, and Z-distance d′Z ≥ d under the stated construction assumptions.Source: B Generator Coefficient Framework for… · pp. 28, 30
• An equivalent CSS code can support the same physical UZ operation as a non-degenerate stabilizer code, with the same generator coefficients and induced logical operator.Source: B Generator Coefficient Framework for… · p. 30

C.1 Proof of Lemma 4

• The Lemma 4 proof applies MacWilliams identities to relate the relevant coset weight enumerators.Source: §C.1 · p. 31

• For the trivial syndrome, the logical operator is RZ(θL) when Aµ=0,γ=0 is real, and ZLRZ(θL) when Aµ=0,γ≠0 is real.Source: §C.3 · p. 32

C.4 Proof of Theorem 6

• The proof obtains ϵ(0,0) = 1 when η = 0 and ϵ(0,0) = 0 when η ≠ 0, using unitarity of UZ.Source: §C.4 · pp. 32–33

C.5 Derivation of (88)

• The derivation of (88) uses that ϵ(a,0)E(a,0) is an element of S.Source: §C.5 · p. 33

C.6 Proof of Theorem 7

• Conversely, assuming (98) makes the nonzero-index coefficients vanish, yielding UZΠS = ΠSUZ.The proof invokes Theorem 6 and the commutation relation for E(0, u).Source: §C.6 · p. 33
• Commutation of UZ with the code-space projector implies the coefficient-vanishing condition (98).The proof uses orthogonality of the syndrome projectors and linear independence of Pauli matrices.Source: §C.6 · p. 34

C.7 Proof of Lemma 11

• For Lemma 11, the pairwise condition (101) implies (105) and (106), and reversing the displayed steps proves the converse.The proof reduces the condition to the divisibility relation in (179) for representatives written using u1, u2, w, and y.Source: §C.7 · p. 34

C.8 Proof of Theorem 12

• The proof follows the idea of Theorem 9 and sets UZ = RZ.It then simplifies (56) using (98).Source: §C.8 · p. 34
• After the simplification, every term in the double sum equals 1, completing the proof.Source: §C.8 · p. 35

C.9 Proof of Lemma 17

• The expression in (186) equals 1 exactly when sw = 1 for every w in C2, equivalent to (129) for all nonzero w in C2.The proof notes s0 = 1.Source: §C.9 · p. 35

C.10 Proof of Lemma 18

• For either case v ∈ Bw or v ∈ Zw \ Bw, the parity calculation places w ∗ (z ⊕ y) ⊕ y′ in the stated projection set.The cases use even parity for both terms in Bw and odd parity for both terms in Zw \ Bw.Source: §C.10 · pp. 35–36
• The proof then reduces completion to showing that |Dw| equals the size of the projection set.Source: §C.10 · p. 36, §C.11 · p. 36

C.11 Proof of Lemma 19

• The proof of Lemma 19 applies the MacWilliams identities to P2θ[Bw] and P2θ[Zw].The passage identifies Pθ[C] as defined in (4) for a linear code C.Source: §C.11 · p. 36