Alexey Gorshkov

Geometric locality is an important theoretical and practical factor for quantum low-density parity-check (qLDPC) codes which affects code performance and ease of physical realization. For device architectures restricted to 2D local gates, naively implementing the high-rate codes suitable for low-overhead fault-tolerant quantum computing incurs prohibitive overhead. In this work, we present an error correction protocol built on a bilayer architecture that aims to reduce operational overheads when restricted to 2D local gates by measuring some generators less frequently than others. We investigate the family of bivariate bicycle qLDPC codes and show that they are well suited for a parallel syndrome measurement scheme using fast routing with local operations and classical communication (LOCC). Through circuit-level simulations, we find that in some parameter regimes bivariate bicycle codes implemented with this protocol have logical error rates comparable to the surface code while using fewer physical qubits.

Entanglement renormalization circuits are quantum circuits that can be used to prepare large-scale entangled states. For years, it has remained a mystery whether there exist scale-invariant entanglement renormalization circuits for chiral topological order. In this paper, we solve this problem by demonstrating entanglement renormalization circuits for a wide class of chiral topologically ordered states, including a state sharing the same topological properties as Laughlin's bosonic fractional quantum Hall state at filling fraction 1/4 and eight states with Ising-like non-Abelian fusion rules. The key idea is to build entanglement renormalization circuits by interleaving the conventional multi-scale entanglement renormalization ansatz (MERA) circuit (made of spatially local gates) with quasi-local evolution. Given the miraculous power of this circuit to prepare a wide range of chiral topologically ordered states, we refer to these circuits as MERA with quasi-local evolution (MERAQLE).

Abstract We present an optimal protocol for encoding an unknown qubit state into a multiqubit Greenberger-Horne-Zeilinger-like state and, consequently, transferring quantum information in large systems exhibiting power-law ($1/r^\alpha$) interactions. For all power-law exponents $\alpha$ between $d$ and $2d+1$, where $d$ is the dimension of the system, the protocol yields a polynomial speedup for $\alpha>2d$ and a superpolynomial speedup for $\alpha\leq 2d$, compared to the state of the art. For all $\alpha>d$, the protocol saturates the Lieb-Robinson bounds (up to subpolynomial corrections), thereby establishing the optimality of the protocol and the tightness of the bounds in this regime. The protocol has a wide range of applications, including in quantum sensing, quantum computing, and preparation of topologically ordered states.

Abstract The field of quantum Hamiltonian complexity lies at the intersection of quantum many-body physics and computational complexity theory, with deep implications to both fields. The main object of study is the LocalHamiltonian problem, which is concerned with estimating the ground-state energy of a local Hamiltonian and is complete for the class QMA, a quantum generalization of the class NP. A major challenge in the field is to understand the complexity of the LocalHamiltonian problem in more physically natural parameter regimes. One crucial parameter in understanding the ground space of any Hamiltonian in many-body physics is the spectral gap, which is the difference between the smallest two eigenvalues. Despite its importance in quantum many-body physics, the role played by the spectral gap in the complexity of the LocalHamiltonian is less well-understood. In this work, we make progress on this question by considering the precise regime, in which one estimates the ground-state energy to within inverse exponential precision. Computing ground-state energies precisely is a task that is important for quantum chemistry and quantum many-body physics. In the setting of inverse-exponential precision, there is a surprising result that the complexity of LocalHamiltonian is magnified from QMA to PSPACE, the class of problems solvable in polynomial space. We clarify the reason behind this boost in complexity. Specifically, we show that the full complexity of the high precision case only comes about when the spectral gap is exponentially small. As a consequence of the proof techniques developed to show our results, we uncover important implications for the representability and circuit complexity of ground states of local Hamiltonians, the theory of uniqueness of quantum witnesses, and techniques for the amplification of quantum witnesses in the presence of postselection.

Quantum Annealing and the Quantum Approximate Optimization Algorithm (QAOA) are both instances of the same control problem in which a combination of two Hamiltonians is applied to minimize the energy of a quantum state. Previous work has suggested that the bang-bang structure of QAOA is optimal but with significant caveats, leaving open the question of what procedure is optimal in practice. In this work, we formalize the optimal control arguments proving that a time-constrained procedure has a bang at the beginning and end but can take on an annealing-like form in-between. In numerics on transverse field Ising models, we show that bang-anneal-bang procedures are common with optimal time-constrained QAOA Trotterizing the annealing portion. We furthermore show an equivalence between different types of time-constraints in this style of problem. The optimal procedures we find are far removed from a monotonically changing adiabatic path in the short-time limit, but we show that as the allowed time increases, the adiabatic limit and intuition from adiabatic quantum computing apply.

Quantum sensing is a promising technology capable of demonstrating clear advantage over comparable classical techniques for precise measurement. One application of quantum sensing is in function estimation, which can be done using a network of entangled quantum sensors, allowing for measurements with greater optimal sensitivity than unentangled sensing protocols. Since quantum sensor networks will likely be used to measure data that should remain private (e.g., biomedical data), it is imperative that these protocols include a cryptographic mechanism to hide sensitive information. In this work, we show that entangled sensor networks are vulnerable to differential attacks. To mitigate these attacks, we introduce secure sensing protocols based on differential privacy. We reconcile Heisenberg-limited scaling and differential privacy and introduce several protocols achieving varying balances between the two. We show that our protocols are resilient to attacks by quantum adversaries and we find advantages in the privacy-utility trade-off when using quantum resources.