quantum hardware designers

Abstract: Decoherence free subspaces are a steady-state structure of the open quantum system which preserves quantum coherence between the states lying with in it and thus has found a variety of applications throughout quantum information science and technology. In this paper we study the computational complexity of deciding whether an open quantum system admits a decoherence free subspace or not. More specifically we study this problem with in the context of Markovian open quantum systems, governed by the time-independent Lindblad master equation. Along the way we introduce the $k$-Local Lindbladian problem, which captures the difficulty of computing purity decay rates under Lindbladian dynamics. We show that both problems are hard for the complexity class Quantum Merlin Arthur (QMA) when the locality $k \geq 5$, with the first under perfect completeness and the second being complete for QMA. Our hardness construction generalizes Kitaev's clock Hamiltonian construction to the open quantum system setting by encoding the execution of a quantum circuit into the steady subspace of a Lindbladian containing both pure and mixed history states. This subspace is then mixed depending on the output of the encoded circuit. Our results suggest that deciding whether a generic Markovian open quantum system admits a decoherence free subspace is intractable even for quantum computation.

Abstract: We prove that deciding the Berry phase for parameterised 2-local qubit Hamiltonians is BQP-complete when presented with a classical description of a guiding state, promised to overlap with the ground state of the system. Our results extend to systems with weighted Heisenberg interactions and when restricted to a 2D square or triangular lattice geometry. The techniques we develop leverage the Schrieffer--Wolff transformation, typically used in the construction of perturbative gadget reductions for local Hamiltonian problems, extending it to parameterised families of Hamiltonians. We demonstrate that there exists a choice of parameterised simulator Hamiltonians whose Berry phase well-approximates that of a parameterised target family. Using the perturbative gadget reduction framework of Oliveira and Terhal and of Schuch and Verstraete, we adapt the arguments to parameterised interactions and demonstrate the error bounds in the resulting simulation can be controlled. Additionally, we provide an explicit proof that families of 1-local Hamiltonians have a Berry phase that can be efficiently computed to inverse-polynomial precision. This establishes a complexity transition between 1-local and 2-local Hamiltonian families.

Abstract: Barren plateau diagnostics characterize whether gradient signal remains available for training, but surviving signal need not translate into successful optimization. We study this trainability--optimization gap at the level of optimizer steps. Treating coefficient-weighted Hamiltonian-term gradients as task-like components, we introduce step-level diagnostics and derive an exact bridge between signed termwise organization, directional activity, and first-order descent. Resolving this bridge into standard first-order geometry shows that the apparent organization--activity factors are not independent optimization axes and that, at fixed state and update norm, the raw gradient maximizes first-order descent of the summed objective. We compare vanilla gradient descent, a deterministic Hamiltonian-term PCGrad variant, and probe-gated LSO-PCGrad on transverse-field Ising model instances with hardware-efficient and Hamiltonian variational ansatzes, together with matched controls for update norm and probe budget. Blind projection can improve an organization diagnostic while worsening final energy and first-order predictability. After conditioning on standard first-order geometry, residual term-space composition shows no reproducible material incremental association with realized descent, while optimizer-relative update norm shows positive material associations in some settings without cross-regime reproducibility. Matched controls provide no resolved final-energy benefit attributable to the projected direction, and the improvement of LSO-PCGrad is more consistent with probe-based search and step-norm adaptation than with Hamiltonian-term projection itself. These results show that gradient-structure diagnostics can characterize trainability and update geometry without serving as standalone evidence of optimization benefit, which requires controls matched on update norm and search budget.

Abstract: Computing dynamical expectation values typically relies on approximations optimized for the state or observable separately, without accounting for how their errors combine in the final expression.This work explores an alternative approach in which each approximation is guided by both the state and the observable, accounting for how they jointly determine the target expectation value. This idea is implemented through a sweep algorithm with coupled loss functions for forward state and backward observable updates. Exact error relations provide an analytical rationale for how the proposed losses can improve the accuracy. Numerical tests on 30-qubit random circuits show errors two to three orders of magnitude smaller than those of variational state compression at equal bond dimensions. These results motivate further exploration of joint state and observable approximation for dynamical expectation value estimation.

Abstract: This paper reports the proof of a strong converse for the unconstrained quantum capacity of the pure-loss bosonic channel. At every fixed rate above capacity, the entanglement-generation fidelity of every code is bounded by a constant times the reciprocal of the number of channel uses. The bound holds without an energy constraint and for arbitrary encoded states, including states correlated across all input modes, and arbitrary joint decoders. The proof combines quantum Chebyshev and hockey-stick testing inequalities with a uniform relative-entropy-variance bound for the balanced pure-loss channel, corresponding to transmissivity $η=1/2$. The variance bound follows by expressing the balanced beam splitter in bright and dark modes: the dark modes are exactly in vacuum, and any state orthogonal to that vacuum contains at least one dark photon. For general transmissivity, dilating the degrading attenuator reduces the problem to this balanced-channel setting and bounds the decoder test by precisely the factor that produces the known quantum-capacity threshold. The resulting argument establishes the strong converse at the unconstrained quantum capacity for every pure-loss bosonic channel.

Abstract: We show that 2-fold Forrelation with inverse-polylogarithmic promise gap can be solved, with bounded error, by polynomial-size QAC$^0$ circuits. Unlike the standard oracle-based Forrelation algorithm, our circuits receive the input explicitly, in the same form as the AC$^0$ circuits against which Forrelation is known to be hard. At constant gap, this yields a natural promise-problem separation between QAC$^0$ and AC$^0$.