Measurement-induced non-commutativity in adaptive fermionic linear optics

Abstract

Fermionic linear optics (FLO) with Gaussian resources is efficiently classically simulable. We show that this is no longer the case for such quantum circuits for fermions with internal degrees of freedom, equipped with mid-circuit number monitoring and classical feedforward. In our architecture, the measurement record routes the selected blocks into a fixed-order Bell-fusion pairing geometry. On the level of classical description, this implies realizing a situation in which the permutation sum no longer collapses to a single determinant or Pfaffian. Each post-selected branch expands as a signed sum of path-ordered products of typically non-commuting dressed blocks, and branch amplitudes are matrix elements of the resulting non-commutative trace polynomials. Numerically, we observe Porter-Thomas statistics as the output distribution and a rapid growth of the minimal order-respecting matrix product operator bond dimension. These results thus establish mid-circuit measurement-induced non-commutativity as a route to sampling hardness for noninteracting fermions under reasonable complexity assumptions, without introducing coherent two-body interactions into the FLO evolution.

Figures (7)

• Figure 1: Monitored adaptive FLO architecture. An auxiliary qudit chain $(A_0,\dots,A_m)$ is coupled to logical qudits $(Q_1,\dots,Q_m)$ and local auxiliary registers $(\widetilde{Q}_1,\dots,\widetilde{Q}_m)$, with each pair $(Q_j,\widetilde{Q}_j)$ encoding the vacuum-plus-single-particle sector of $d$ fermionic orbitals per block. Occupied input blocks $\mathcal{I}_{\rm in}=\{m{-}n{+}1,\dots,m\}$ are Bell-entangled with the auxiliary chain. A number-conserving FLO circuit (blue) with propagator $\mathbf V$ acts on the encoded orbitals. Mid-circuit monitoring (MCM) produces $\boldsymbol c$; collision-free outcomes select $n$ blocks, fix the $n\times n$ block sub-matrix $\mathbf S$, and route them via feedforward into a fixed nearest-neighbor Bell-fusion geometry with outcomes $\boldsymbol\beta$, followed by a boundary readout on the terminal auxiliary. For fixed boundary state vectors $|\ell\rangle$ and $|r\rangle$, the protocol defines the conditional branch distribution $p(\boldsymbol\beta\,|\,\boldsymbol c)$.
• Figure 2: Collision-free post-selection probability $p_{\rm e}(n,m,d)$ in the dilute regime ($m=\kappa n^2$). Empirical distributions of per-instance rates (over 200 Haar-random realizations) are compared against the asymptotic prediction $p_\infty=\exp[-(1-1/d)/(2\kappa)]$ (dashed lines, Lemma \ref{['lem:post-selection-rate']}). (a) $d=2$ and (b) $d=3$, both with $\kappa\in\{1/2,1/4\}$.
• Figure 3: Order-respecting contraction memory and non-commutativity diagnostics for monitored-FLO branches. (a, b) Truncated minimal MPO bond dimension $\chi_{\max}^{(\varepsilon)}$ ($\varepsilon=10^{-3}$) in the enforced fusion order (red: monitored-FLO; blue: commuting control with $S_{t,k}\propto \mathbbm 1_d$ and $Z$-only byproducts; dash-dotted: exponential fit). (c, d) Normalized commutator diagnostic $\nu_{\mathrm{nc}}$ of the dressed blocks [Eq. \ref{['eq:nc-score-def']}]; dotted line: i.i.d. Ginibre reference mean. Panels (c, d) are shown for larger $n$ to highlight the concentration of $\nu_{\mathrm{nc}}$ with system size. (a, c) $d=2$. (b, d) $d=3$. Statistics are computed over 200 independent instances per $n$.
• Figure 4: Second-moment ($\Gamma(\boldsymbol c)=\sum_{\boldsymbol\beta}p(\boldsymbol\beta\,|\,\boldsymbol c)^2$) and Porter-Thomas diagnostics of monitored-FLO branches. (a, b) Haar-normalized second moment versus $n$. Red: monitored-FLO ensemble. Blue: commuting-control benchmark ($S_{t,k}\propto \mathbbm 1_d$ with $Z$-only byproducts). The dashed line marks the Haar random-state average. Statistics are computed over 200 independent instances per $n$. (c, d) Distributions of rescaled branch probabilities $x:=d^{2n}p(\boldsymbol\beta\,|\,\boldsymbol c)$ for representative instances with $n=8$, compared to the Porter-Thomas law $P(x)=e^{-x}$ (grey dashed). (a, c) $d=2$. (b, d) $d=3$.
• Figure 5: Local $d=2$ encode/decode gadgets on the pair $(Q_j,f_j)$. Left: the encoder $\mathcal{E}_j$ maps $|\alpha\rangle_{Q_j}|0\rangle_{f_j}$ to the one-hot codewords $|1,0\rangle$ ($\alpha=0$) and $|0,1\rangle$ ($\alpha=1$). Right: the decoder $\mathcal{D}_j$ applies a local two-qubit gate and measures $f_j$, producing the coarse-grained occupation flag $c_j$ while leaving the internal state on $Q_j$ unresolved.

Theorems & Definitions (17)

• Lemma 1: Constant-rate collision-free monitoring
• Lemma 2: FLO-induced determinantal kernel
• Proposition 1: Non-commutative branch structure
• Theorem 1: Exponential sequential-memory barrier
• Conjecture 1: Average-case hardness of monitored-FLO branches
• Theorem 2: Conditional hardness of approximate sampling
• proof : Proof of Lemma \ref{['lem:post-selection-rate']}
• proof : Proof of Lemma \ref{['lem:ftpd']}
• Lemma 3: Three-party Bell-projection identity
• proof