The limits of magic-state distillation

The paper in detail

Fault-tolerant computers make Clifford operations cheap, but universality needs magic states, usually distilled from many noisy ones. Each input is \(\rho_T(p)=(1-p)\ketbra TT+pZ\ketbra TTZ\) with \(\ket T=(\ket0+e^{i\pi/4}\ket1)/\sqrt2\). A fault pattern \(e\in\mathbb F_2^n\) marks the faulty inputs, giving orthonormal pure inputs \(\ket{\phi_e}=Z^e\ket T^{\otimes n}\). A protocol has suppression order \(d\) when it accepts the fault-free input and every pattern with fewer than \(d\) faults is harmless, meaning it yields exactly the target whenever accepted. Earlier general bounds gave little beyond \(n\gtrsim d\).

Theorem 1: some stabilizer protocol tolerates a set \(H\) of patterns if and only if some stabilizer code \(C\) on the inputs, with Clifford encoding \(U\), satisfies \(\Pi_C\ket{\phi_0}\ne0\) and \(\Pi_C\ket{\phi_e}\in\mathbb C\,U\ket\tau\) for all \(e\in H\). Every tolerated input collapses onto the encoded target or is rejected. Theorem 2: flipping the sign of \(\sqrt2\) commutes with stabilizer operations, fixes targets with rational Pauli expectations such as CCZ, and sends \(\rho_T(p)\) to \(\rho_T(1-p)\). A pattern is therefore harmless exactly when its complement is, so patterns with \(|e|>n-d\) must be harmless too.

These reductions turn an unbounded space of protocols into a search over signed stabilizer codes, up to \(9\times10^{11}\) of them, tested in exact arithmetic. Results (Table 1): \(n_{\min}(T,2)=9\), \(n_{\min}(\mathrm{CS},2)=8\), \(n_{\min}(\mathrm{CCZ},2)=8\), and exact values for several first-order targets, for example six T states for CCCZ. Analytic witnesses give \(n\ge2d+1\) for rational non-stabilizer targets and \(n\ge2d\) for one T state at every order. A two-level design gives a quartic CCZ factory with 48 inputs and leading error \(864p^4\), under a third of the previous best.

The new T factory measures eight explicit 9-qubit stabilizers, postselects on all outcomes \(+1\) (probability \(1/128\) at \(p=0\)) and decodes by a Clifford, with output error \(\tfrac{27}2p^2+O(p^3)\). It lies outside the generalized triorthogonal families. The authors identify three routes to closing the remaining gaps: proving that collapsing codes contain \(Z^{\otimes n}\), stronger convex relaxations, and richer multi-level designs.

Transferable methods

1. Reduce all protocols to one code projection

   Because damage is nonnegative, every accepted branch of a good protocol must itself be good, and each branch is a code projection followed by a Clifford; tolerating faults becomes a geometric collapse condition.

   Collapse for a code \(C\): with Gram matrix \(G_{ef}=\bra{\phi_e}\Pi_C\ket{\phi_f}\), pattern \(e\) is tolerated when \(G_{00}G_{ee}=|G_{0e}|^2\), so \(\Pi_C\ket{\phi_e}\) is parallel to \(\Pi_C\ket{\phi_0}\), or when \(G_{ee}=G_{0e}=0\), so it is rejected. Worked example: measuring \(Z_1Z_2Z_3\) on \(\ket T^{\otimes3}\) succeeds with probability \(\tfrac12\). The all-faulty pattern collapses onto the fault-free vector because \(Z^{\otimes3}\) is a stabilizer, while a single fault acts as a logical Pauli and is harmful, so the order is one.

   Editorial verification of Proposition 7. The eight listed stabilizers commute; projecting \(\ket T^{\otimes9}\) onto the code gives acceptance probability exactly \(1/128\). For each of the nine single faults, \(\Pi_CZ_i\ket T^{\otimes9}\) is either zero or parallel to \(\Pi_C\ket T^{\otimes9}\). The identity of Proposition 3, \(\mathrm{CS}_{12}\mathrm{CS}_{13}\ket{{+}{+}{+}}=\mathrm{CX}_{23}H_2\mathrm{CX}_{12}S_2H_2\ket{\mathrm{CCZ}}\), also holds, with overlap 1.

   How to reuse it

   When optimizing over adaptive, postselected protocols with a nonnegative failure measure, check whether each accepted branch must already succeed. If so, the optimum is attained by one branch, which often has a rigid normal form such as a code projection, Kraus operator or measurement, and the search becomes finite.

2. Galois conjugation pairs complementary faults

   The field automorphism \(\sqrt2\mapsto-\sqrt2\) maps noisy T states with error \(p\) to error \(1-p\) while fixing every stabilizer operation and every rational target, which doubles the set of patterns a protocol must tolerate.

   \(\rho_T(p)=\tfrac12\bigl[I+\tfrac{1-2p}{\sqrt2}(X+Y)\bigr]\), so \(\sigma:\sqrt2\mapsto-\sqrt2\) sends \(\rho_T(p)\) to \(\rho_T(1-p)\), and hence \(\rho_e\) to \(\rho_{e+\mathbf1}\). Stabilizer branches act on Pauli coefficients by rational matrices, so they commute with \(\sigma\). For a target with rational Pauli expectations, the damages of \(e\) and \(e+\mathbf1\) are \(\sigma\)-conjugates in \(\mathbb Q(\sqrt2)\), and one vanishes exactly when the other does.

   Consequence (editorial phrasing). With order \(d\), all patterns with \(|e|<d\) and all with \(|e|>n-d\) must be harmless. For small \(n\) this forces a witness state, built only from harmless patterns, to be converted into the target by stabilizer operations, which is impossible. This gives \(n\ge2d+1\) for rational targets that are not stabilizer states.

   How to reuse it

   When inputs and allowed operations have entries in a number field, look for field automorphisms that commute with the operations. They map one physical situation to another, here \(p\leftrightarrow1-p\), and any property preserved by the automorphism must hold in both, often halving a search or yielding a lower bound.

Source: Section II (Theorems 1 and 2, worked example, Proposition 3); Section III (Theorem 4, Propositions 5 and 6); Section IV (Proposition 7, two-level factories). Based on the linked paper version. Derivations marked as editorial, and suggested uses in other problems, are editorial guidance rather than results of the paper.

Context & limitations

Why it matters. First lower bounds that survive arbitrary postselection and adaptivity while depending strongly on \(d\). They settle the practically relevant quadratic regime, since cultivated input states already have small error.

Limitations. Inputs are counted per run with ideal Clifford operations, not space-time cost; per accepted output the 9-input factory consumes more inputs than the earlier 10-input protocol, since it succeeds with probability 1/128. Cubic and quartic cases keep gaps (one T state: 10–15 and 11–48). The searches are computer-assisted, and the authors report using a language model for assistance.

Description based on the full text of the linked arXiv version, read 2026-10-23; equations follow the paper’s notation.