Quantum computing · Theory and numerical analysis
Correlation-sensitive magic-state distillation from spin-squeezed neutral atoms
arXiv:2610.36963v1 ↗PDF ↗Submitted 27 October 2026
- Question
- Magic-state distillation is analysed for independent noisy inputs, with a fidelity threshold above which each round improves the resource. Neutral-atom factories prepare many resource blocks in parallel under shared interactions and control fields, which correlates their errors. Can such correlations defeat distillation even when every input is above threshold, and which measurable quantity decides?
- Result
- Yes. For the five-to-one Bravyi–Kitaev protocol with independent local Clifford twirls, the accepted output depends on the joint input only through sums \(A_k\) of \(k\)-body correlators along the target axis: \(p_{\rm acc}=(3+A_4)/48\) and \(t_{\rm out}=(A_3-2A_5)/(3+A_4)\). Distillation improves on the mean input exactly when \(G=A_3-2A_5-\bar t(3+A_4)>0\). Inputs with identical one- and two-qubit reduced states can have opposite gains, and in finite-range Ising simulations of a 25-atom source, interblock correlations reverse the gain while the product of the same local states gains.
The paper in detail
Each five-atom block is projected into the \([[5,1,3]]\) code, generated by \(XZZXI\), \(IXZZX\), \(XIXZZ\) and \(ZXIXZ\), and five accepted logical qubits enter the outer Bravyi–Kitaev step, whose target \(\ket{T_{\rm BK}}\) has Bloch vector \((1,1,1)/\sqrt3\). For any joint input, \(p_{\rm acc}=\operatorname{Tr}(\rho\Pi)\) and the logical Bloch components are stabilizer expectation values divided by \(p_{\rm acc}\). Independently applying a random cyclic Clifford to each input dephases it in the eigenbasis of \(M=(X+Y+Z)/\sqrt3\) but keeps the joint distribution \(P(\mathbf s)\) of the target-axis signs, summarized by \(A_k=\sum_{|S|=k}\langle\prod_{i\in S}s_i\rangle\).
Stabilizers have weight four and logical operators weight three and five, so acceptance depends on fourth-order correlations and output polarization on the competition between third- and fifth-order ones. For independent identical inputs \(A_k=\binom5kt^k\), and \[t^{\rm iid}_{\rm out}-t=\frac{t(1-t^2)(7t^2-3)}{3+5t^4}\] gives the familiar threshold \(F_{\rm th}=(1+\sqrt{3/7})/2\approx0.827\). A family with fixed single-qubit and pair states moves from loss to gain as only the higher-order statistics change, and a GHZ-type family \(\alpha^2\rho_+^{\otimes5}+\beta^2\rho_-^{\otimes5}\) is accepted with probability \(1/6\) and output fidelity equal to its input fidelity \(\alpha^2\).
As a physical source, each block starts in \(\ket+^{\otimes5}\) and evolves under finite-range Ising couplings, which produce a spin-squeezed, nonstabilizer precursor. Projection alone wastes it: \(X^{\otimes5}\) is a logical operator that commutes with the dynamics and the code, so the accepted state is always \(\ket+_L\). Addressed \(S\) gates on two sites turn this parity into \(Y_1Y_2X_3X_4X_5\), which anticommutes with every generator, and the projected state then reaches near-unit target fidelity after a logical Clifford alignment.
For 25 atoms in five blocks with couplings \(J(r)=[1+(r/R)^6]^{-1}\), comparing the joint state with the product of its own reduced states isolates the effect of correlations: in a finite region every input exceeds \(F_{\rm th}\), the correlated register loses fidelity, and the product register gains. Larger gaps between blocks suppress interblock coupling while keeping the intrablock interaction. A protocol that stores accepted source outputs, with syndrome-readout confusion, atom loss and storage dephasing, reaches positive gain above a threshold source contrast. The sign of \(G\) can be estimated from target-axis readout of accepted registers, with a sample cost growing as \(G^{-2}\) near the boundary.
Transferable methods
Stabilizer expansion of a code projection on correlated inputs
After independent twirls each input is diagonal in the target basis, and every stabilizer or logical Pauli string picks out the correlator of the qubits it acts on. Acceptance and output polarization therefore become linear in correlation sums of fixed orders, given by the weights in the stabilizer group and the logical cosets.
With \(\Pi=\tfrac1{16}\sum_{G\in\mathcal S}G\), \(p_{\rm acc}=\operatorname{Tr}(\rho\Pi)\) and \(r_a=\operatorname{Tr}(\rho\Pi\bar\sigma_a)/p_{\rm acc}\). For \(\rho_s=(I+sM)/2\) every single-qubit Pauli has \(\operatorname{Tr}(P\rho_s)=s/\sqrt3\), so a weight-\(w\) string contributes a multiple of a \(w\)-body correlator. The 15 nontrivial stabilizers have weight four and the logical cosets weight three and five, which gives \[p_{\rm acc}=\frac{3+A_4}{48},\qquad t_{\rm out}=\frac{A_3-2A_5}{3+A_4}.\]
Editorial check: projecting random correlated sign distributions on five qubits onto the code space numerically, and applying \(XS\) to the logical Bloch vector, reproduces both formulas to machine precision in every trial (for example \(p_{\rm acc}=0.06046\), \(t_{\rm out}=0.2084\)). For the twirled GHZ-type family, \(A_k=\binom5k\) for even \(k\) and \(\binom5k(\alpha^2-\beta^2)\) for odd \(k\), so \(p_{\rm acc}=8/48=1/6\) and \(t_{\rm out}=\alpha^2-\beta^2\), that is, \(F_{\rm out}=\alpha^2\).
How to reuse it
Any stabilizer-code postselection acting on twirled, correlated inputs is a linear function of correlator sums whose orders are the weights in the stabilizer group and the logical cosets. Reading off those weights tells you which correlations affect yield and which affect quality, and gives a gain criterion that needs no tomography.
Symmetry selection under code projection
If the preparation unitary and the code projector share a symmetry that is a logical operator, the projected state is pinned to that operator’s eigenstate, whatever nonstabilizer structure existed before projection. Local Clifford gates that turn the symmetry into an operator anticommuting with the code remove the constraint.
\(L=X^{\otimes5}=\bar X\) commutes with \(U_{ZZ}=\exp[-i\sum_{i<j}\theta_{ij}Z_iZ_j]\) and with \(\Pi\), and \(L\ket+^{\otimes5}=\ket+^{\otimes5}\), so \[L\,\Pi U_{ZZ}\ket+^{\otimes5}=\Pi U_{ZZ}\ket+^{\otimes5}:\] the accepted state is always \(\ket+_L\), a stabilizer state, for any Ising couplings and also for one-axis twisting, since \(J_z^2=\tfrac54I+\tfrac12\sum_{i<j}Z_iZ_j\).
After \(S_1S_2\), using \(SXS^\dagger=Y\), the conserved parity becomes \(W=Y_1Y_2X_3X_4X_5\). Editorial check: \(W\) differs from each generator on an odd number of non-identity sites, so it anticommutes with all four, \(\Pi W\Pi=0\), and the projection no longer fixes a logical eigenvalue; \(X^{\otimes5}\) commutes with all four. The Ising evolution supplies the non-Clifford resource, and the phases, projection and alignment \(A_L=Y_LH_L\) are stabilizer operations that release it.
How to reuse it
When code projection of an interacting state returns only stabilizer states, look for a symmetry shared by the dynamics and the code; addressed Clifford gates that map it to an operator anticommuting with the stabilizers unlock the resource. Conversely, such shared symmetries explain why some physically rich states yield nothing useful after projection.
Context & limitations
Why it matters. Pair characterization of a resource register does not predict distillation; the paper gives an exact criterion in terms of moments that local target-axis readout measures, with finite-sample confidence bounds. It also shows how to obtain encoded magic from a spin-squeezed Ising precursor: addressed \(S\) gates break the global parity that otherwise forces the projected output to a stabilizer state, and spacing between blocks suppresses the harmful interblock correlations.
Limitations. The exact map assumes independent local twirls and the five-to-one protocol on the \([[5,1,3]]\) code; other protocols need their own expansion. The neutral-atom results are simulations of a one-dimensional soft-core interaction model with component fidelities taken from other experiments. Direct preparation of the input state uses fewer atoms when it already reaches the required quality, so the squeezed route pays off only where the Ising interaction is the cheaper resource.