28 Lem 6: Spacetime Decoupling
This chapter establishes the spacetime decoupling lemma: any spacetime logical fault \(F\) is equivalent, up to multiplication by spacetime stabilizers, to the product of a pure space logical fault and a pure time logical fault:
where \(F_{\mathrm{space}}\) consists only of Pauli errors at a single time step, and \(F_{\mathrm{time}}\) consists only of measurement/initialization errors.
A spacetime fault \(F\) is a pure space fault at a single time step \(t\) if:
All space errors occur only at time \(t\): for all qubits \(q\) and times \(t' \neq t\), the space error \(F.\mathrm{spaceErrors}(q, t') = I\).
There are no time errors: for all measurements \(m\) and times \(t'\), \(F.\mathrm{timeErrors}(m, t') = \mathrm{false}\).
A spacetime fault \(F\) is a pure time fault (only measurement/initialization errors) if for all qubits \(q\) and times \(t\),
Two spacetime faults \(F\) and \(G\) are equivalent modulo stabilizers (with respect to a detector collection \(\mathrm{DC}\), base outcomes, and logical effect predicate) if there exists a spacetime stabilizer \(S\) such that \(F = G \cdot S\).
A gauging interval \([t_i, t_o]\) consists of:
An initial boundary time step \(t_i\).
A final boundary time step \(t_o\).
The ordering constraint \(t_i {\lt} t_o\).
A Pauli pair move captures a single operation of moving a Pauli error from time \(t\) to \(t+1\) using a Pauli pair stabilizer. It consists of:
A qubit location (vertex or edge).
The source time step (fromTime).
The Pauli type (\(X\), \(Y\), or \(Z\)).
The induced measurement faults on anticommuting checks.
A cleaning sequence is a list of Pauli pair moves that collectively move all Pauli errors to a target time \(t_{\mathrm{ref}}\). The combined effect is:
Space errors: net effect is to relocate all Paulis to \(t_{\mathrm{ref}}\).
Time errors: XOR of all induced measurement faults.
The combined measurement errors of a cleaning sequence is the function \(M \to \mathrm{TimeStep} \to \mathrm{Bool}\) obtained by folding over the list of moves: for each measurement \(m\) and time \(t\), the result is the XOR of all induced measurement faults from moves whose source time or successor time equals \(t\).
Let \(\mathrm{DC}\) be a detector collection, and suppose the logical effect predicate is group-like and the syndrome is a group homomorphism. If \(S\) is a spacetime stabilizer, then \(S^{-1}\) is also a spacetime stabilizer.
We must show that \(S^{-1}\) has empty syndrome and preserves the logical information.
Empty syndrome for \(S^{-1}\): This follows from the fact that the syndrome is a group homomorphism, so it respects inverses. Since the syndrome of \(S\) is empty, the syndrome of \(S^{-1}\) is also empty.
Preserves logical for \(S^{-1}\): This follows from the group-like property of the logical effect. Since \(S\) preserves the logical information, so does \(S^{-1}\).
Let \(\mathrm{DC}\) be a detector collection, let the logical effect be group-like, and let the syndrome be a group homomorphism. Let \(I = [t_i, t_o]\) be a gauging interval, and let \(F\) be a spacetime logical fault.
Suppose that for any such fault, there exists a cleaning stabilizer \(S_{\mathrm{clean}}\) (built from Pauli pair stabilizers as in Lemma 4) such that \(S_{\mathrm{clean}}\) is a spacetime stabilizer and the space errors of \(F \cdot S_{\mathrm{clean}}\) are concentrated at time \(t_i\).
Then there exist spacetime faults \(F_{\mathrm{space}}\) and \(F_{\mathrm{time}}\) such that:
\(F\) is equivalent to \(F_{\mathrm{space}} \cdot F_{\mathrm{time}}\) modulo stabilizers.
\(F_{\mathrm{space}}\) is a pure space fault at the single time step \(t_i\).
\(F_{\mathrm{time}}\) is a pure time fault (only measurement errors).
We extract the cleaning stabilizer \(S_{\mathrm{clean}}\) from the hypothesis, obtaining that \(S_{\mathrm{clean}}\) is a spacetime stabilizer and the space errors of \(F' := F \cdot S_{\mathrm{clean}}\) are concentrated at \(t_i\).
We define the two components:
\(F_{\mathrm{space}}\): the spacetime fault with space errors given by \(F'.\mathrm{spaceErrors}(q, t)\) when \(t = t_i\) and \(I\) otherwise, and with no time errors.
\(F_{\mathrm{time}}\): the spacetime fault with no space errors (all \(I\)) and time errors equal to \(F'.\mathrm{timeErrors}\).
We verify the three claims using the witness \(F_{\mathrm{space}}\), \(F_{\mathrm{time}}\):
Claim 1 (\(F \sim F_{\mathrm{space}} \cdot F_{\mathrm{time}}\)): We use \(S = S_{\mathrm{clean}}^{-1}\) as the stabilizer witness. First, \(S_{\mathrm{clean}}^{-1}\) is a stabilizer by the stabilizer inverse lemma. Second, we must show \(F = (F_{\mathrm{space}} \cdot F_{\mathrm{time}}) \cdot S_{\mathrm{clean}}^{-1}\).
We first establish that \(F' = F_{\mathrm{space}} \cdot F_{\mathrm{time}}\) by extensionality:
For space errors: if \(t = t_i\), then \((F_{\mathrm{space}} \cdot F_{\mathrm{time}}).\mathrm{spaceErrors}(q, t) = F'.\mathrm{spaceErrors}(q, t) \cdot I = F'.\mathrm{spaceErrors}(q, t)\) by simplification using \(P \cdot I = P\). If \(t \neq t_i\), then \((F_{\mathrm{space}} \cdot F_{\mathrm{time}}).\mathrm{spaceErrors}(q, t) = I \cdot I = I\), and by the concentration hypothesis, \(F'.\mathrm{spaceErrors}(q, t) = I\) as well.
For time errors: \((F_{\mathrm{space}} \cdot F_{\mathrm{time}}).\mathrm{timeErrors}(m, t') = \mathrm{false} \oplus F'.\mathrm{timeErrors}(m, t') = F'.\mathrm{timeErrors}(m, t')\) by simplification using \(\mathrm{false} \oplus b = b\).
Then we compute:
using the identity law, the inverse cancellation \(S_{\mathrm{clean}} \cdot S_{\mathrm{clean}}^{-1} = 1\), and associativity of multiplication.
Claim 2 (\(F_{\mathrm{space}}\) is a pure space fault at \(t_i\)): By construction, for any qubit \(q\) and time \(t' \neq t_i\), \(F_{\mathrm{space}}.\mathrm{spaceErrors}(q, t') = I\) by the conditional definition, and for all measurements \(m\) and times \(t'\), \(F_{\mathrm{space}}.\mathrm{timeErrors}(m, t') = \mathrm{false}\) by definition.
Claim 3 (\(F_{\mathrm{time}}\) is a pure time fault): By construction, for all qubits \(q\) and times \(t\), \(F_{\mathrm{time}}.\mathrm{spaceErrors}(q, t) = I\).
Let \(F_{\mathrm{space}}\) and \(F_{\mathrm{time}}\) be spacetime faults, both with empty syndrome. Then each of \(F_{\mathrm{space}}\) and \(F_{\mathrm{time}}\) is either a spacetime stabilizer or a spacetime logical fault.
We apply the empty syndrome partition theorem (which states that any fault with empty syndrome is either a stabilizer or a logical fault) separately to \(F_{\mathrm{space}}\) using the hypothesis \(h_{\mathrm{space\_ syndrome}}\), and to \(F_{\mathrm{time}}\) using the hypothesis \(h_{\mathrm{time\_ syndrome}}\). The result is the conjunction of the two dichotomies.
Let the logical effect be group-like. If \(F_{\mathrm{space}} \cdot F_{\mathrm{time}}\) affects the logical information, \(F_{\mathrm{space}}\) has empty syndrome, and \(F_{\mathrm{time}}\) is a spacetime stabilizer, then \(F_{\mathrm{space}}\) is a spacetime logical fault.
We construct the proof that \(F_{\mathrm{space}}\) is a spacetime logical fault:
Empty syndrome: This is given directly by hypothesis.
Affects logical: We unfold the definition of “affects logical information” and proceed by contradiction. Assume \(F_{\mathrm{space}}\) preserves the logical information. Since \(F_{\mathrm{time}}\) is a stabilizer, it also preserves the logical information. By the group-like property of the logical effect (specifically, that the product of two faults preserving logical also preserves logical), the product \(F_{\mathrm{space}} \cdot F_{\mathrm{time}}\) preserves logical. This contradicts the hypothesis that \(F_{\mathrm{space}} \cdot F_{\mathrm{time}}\) affects the logical information.
Let the logical effect be group-like. If \(F_{\mathrm{space}} \cdot F_{\mathrm{time}}\) affects the logical information, \(F_{\mathrm{time}}\) has empty syndrome, and \(F_{\mathrm{space}}\) is a spacetime stabilizer, then \(F_{\mathrm{time}}\) is a spacetime logical fault.
The argument is symmetric to the previous theorem. We construct the proof that \(F_{\mathrm{time}}\) is a spacetime logical fault:
Empty syndrome: This is given directly by hypothesis.
Affects logical: We unfold the definition of “affects logical information” and proceed by contradiction. Assume \(F_{\mathrm{time}}\) preserves the logical information. Since \(F_{\mathrm{space}}\) is a stabilizer, it also preserves the logical information. By the group-like property of the logical effect, the product \(F_{\mathrm{space}} \cdot F_{\mathrm{time}}\) preserves logical. This contradicts the hypothesis that \(F_{\mathrm{space}} \cdot F_{\mathrm{time}}\) affects the logical information.