Joshua M. Courtney

Joshua M. Courtney

We achieve query-optimal quantum simulations of non-Hermitian Hamiltonians $H_{\mathrm{eff}} = H_R + iH_I$, where $H_R$ is Hermitian and $H_I \succeq 0$, using a bivariate extension of quantum signal processing (QSP) with non-commuting signal operators. The algorithm encodes the interaction-picture Dyson series as a polynomial on the bitorus, implemented through a structured multivariable QSP (M-QSP) circuit. A constant-ratio condition guarantees scalar angle-finding for M-QSP circuits with arbitrary non-commuting signal operators. A degree-preserving sum-of-squares spectral factorization permits scalar complementary polynomials in two variables. Angles are deterministically calculated in a classical precomputation step, running in $\mathcal{O}(d_R \cdot d_I)$ classical operations. Operator norms $α_R\,,β_I$ contribute additively with query complexity $\mathcal{O}((α_R + β_I)T + \log(1/\varepsilon)/\log\log(1/\varepsilon))$ matching an information-theoretic lower bound in the separate-oracle model, where $H_R$ and $H_I$ are accessed through independent block encodings. The postselection success probability is $e^{-2β_I T}\|e^{-iH_{\mathrm{eff}}T}|ψ_0\rangle\|^2\cdot (1 - \mathcal{O}(\varepsilon))$, decomposing into a state-dependent factor $\|e^{-iH_{\mathrm{eff}}T}|ψ_0\rangle\|^2$ from the intrinsic barrier and an $e^{-2β_I T}$ overhead from polynomial block-encoding.

Joshua M. Courtney

Multivariate quantum signal processing (M-QSP) has recently been shown to be applicable for non-Hermitian Hamiltonian simulation, opening several problems regarding the optimization landscape, angle-finding, and constant-factor analysis. We resolve several of these problems here. We find the anti-Hermitian query complexity $d_I = Θ(\betaI T + \log(1/\varepsilon)/\log\log(1/\varepsilon))$ to be tight, established via Chebyshev coefficient bounds, modified Bessel function asymptotics, and Lambert~$W$ inversion. Fast-forwarding to $d_I = \mathcal{O}(\sqrt{\betaI T})$ is impossible in the bivariate polynomial model, though a linear state-dependent improvement to $d_I = \mathcal{O} β_{\mathrm{eff}} T + \log(1/\varepsilon)/\log\log(1/\varepsilon))$ is achievable. The optimization landscape of M-QSP admits spurious local minima, but a warm-start basin guarantee ensures the two-stage algorithm converges. CRC-exploiting block peeling reduces angle-finding from $\mathcal{O}(d^3)$ to $\mathcal{O}(d^2)$ classical operations, and optimized error allocation yields a leading constant of approximately~$2$ relative to the information-theoretic lower bound. A constant-ratio condition extends to non-identical signal operators, enabling time-dependent non-Hermitian simulation with query complexity $\mathcal{O}(\int_0^T(\alphaR(s) + \betaI(s))\,ds + \log(1/\varepsilon)/\log\log(1/\varepsilon))$. Block-encoding overhead $e^{-2\betaI T}$ holds across all function classes within the walk-operator oracle model, and dilational methods (Schrödingerization) achieve the walk-operator barrier. A precisely characterized direct-access construction achieves the intrinsic barrier $e^{-2ωT}$ (with $ω< \betaI$ for non-commuting Hamiltonians) on a restricted domain, though extension to the full bitorus remains open.

Joshua M. Courtney

We consider the routing of neutral atoms on a reconfigurable lattice in terms of hypergraph transformations. We prove the routing number of a Ramanujan $(d,r)$-regular hypergraph on $N$ vertices satisfies $\mathrm{rt}(H) = Θ(\log N)$, where routing is via matchings in the clique expansion graph $G_{\mathrm{cl}}(H)$. Hypergraphs reframe the qubit routing problem by replacing Nenadov's two-sided spectral gap hypothesis with a one-sided condition based on eigenvalue centering. Song--Fan--Miao (SFM) coverings scale for Ramanujan families of every uniformity. A virtual overlay theorem establishes a capacity--depth tradeoff for 3D acousto-optic lens (AOL) architectures, with multi-layer stacking achieving $Θ(\log N)$ routing with $L = O(\log N)$ independent overlay layers. An abelian Alon--Boppana barrier shows that fixed-degree Cayley graphs on $\mathbb{Z}_n^2$ cannot be Ramanujan and affine derandomization on such graphs achieves 15--30% congestion reduction. Towers of $k$-fold Ramanujan coverings yield $\mathrm(H_L) = O(\log N)$ by recursive routing lift. Entanglement-assisted routing by pre-distributed Bell pairs achieves $O(\log N)$ teleportation depth with a stable crossover at $\sim\!4$ routing rounds. Displacement energy analyzes greedy adaptive routing, identifying stalling and a hybrid greedy--Valiant protocol achieving $\sim\!3\times$ speedup at practical scales. Hierarchical multi-scale routing achieves $O(\log^2 N / \log b)$ depth with boundary-only transfers at capacity $k = O(\sqrt{N} \log N)$, and $O(\log N)$ depth with optimal block size $b = Θ(\sqrt{n})$.

Joshua M. Courtney

We analyze permutation routing of rigid blocks representing surface code patches of $d_C^2$ atoms on a reconfigurable lattice with hypergraph transformations. For a hypergraph $H$, code distance $d_C$, $s=d_C^2$, number of blocks $N_L$, and guard distance $g$, we show the block routing number $\mathrm{rt}_B(H, s, g) = Θ(d_C \log N_L)$. A spectral analysis of the quotient graph $Q(G_{\mathrm{cl}}(H), B)$ (blocks as supervertices) shows that the spectral ratio $β_Q < 1$ is preserved in the high-connectivity regime. Negative association of block permutations and congestion bounds are used for random intermediate configurations. Serialization establishes that each quotient routing phase requires $O(d_C)$ physical sub-steps due to the block footprint width. A lower bound $\mathrm{rt}_B = Ω(d_C \log N_L)$ follows from combining the spectral lower bound on quotient phases with the traversal cost per phase. We include error model analysis grounded in recent experimental results, syndrome extraction protocols (stop-and-correct, rolling active fault-tolerant (AFT) measurement, and adaptive deformation), and integration with lattice surgery compilation via the Litinski protocol. Composition with the correlated-decoding scheme reduces syndrome-extraction overhead from $O(d_C)$ to $O(1)$ per correction window, leaving routing as the leading-order contributor to the integrated $O(d_C \log N_L)$ depth. Spectral inheritance is organized in a hierarchy: exact (Haemers interlacing on equitable partitions), perturbative (Weyl bounds for near-equitable partitions, a practically relevant case for surface-code patches), and universal (higher-order Cheeger). Methods extend directly to QCCD trapped-ion architectures under the same regime condition, with junction crossings replacing AOD transports as the elementary single-hop translation.