We study the problem of implementing arbitrary permutations of qubits under interaction constraints in quantum systems that allow for arbitrarily fast local operations and classical communication (LOCC). In particular, we show examples of speedups over swap-based and more general unitary routing methods by distributing entanglement and using LOCC to perform quantum teleportation. We further describe an example of an interaction graph for which teleportation gives a logarithmic speedup in the worst-case routing time over swap-based routing. We also study limits on the speedup afforded by quantum teleportation—showing an O ( N log N ) upper bound on the separation in routing time for any interaction graph—and give tighter bounds for some common classes of graphs. Published by the American Physical Society 2024
We study the problem of implementing arbitrary permutations of qubits under interaction constraints in quantum systems that allow for arbitrarily fast local operations and classical communication (LOCC). In particular, we show examples of speedups over swap-based and more general unitary routing methods by distributing entanglement and using LOCC to perform quantum teleportation. We further describe an example of an interaction graph for which teleportation gives a logarithmic speedup in the worst-case routing time over swap-based routing. We also study limits on the speedup afforded by quantum teleportation—showing an O ( N log N ) upper bound on the separation in routing time for any interaction graph—and give tighter bounds for some common classes of graphs. Published by the American Physical Society 2024
Locality constrains the flow of information between different parts of many-body quantum systems. In quantum computers, this affects the ability to perform arbitrary interactions for quantum information processing tasks. A crucial challenge for scalable quantum architectures is thus to minimize the overheads due to locality constraints. Locality constraints also affect the way information and entanglement can be spread in many body quantum systems, and our ability to make predictions about such systems. One way to implement arbitrary interactions in locality-constrained architectures is by permuting qubits, also known as performing quantum routing. In the first part of this thesis, we investigate quantum routing strategies using emerging architectural capabilities beyond swap gates. We first investigate the use of mid-circuit measurements and classical feedback, which enable quantum teleportation. We prove upper bounds on the speedup afforded by the use of quantum teleportation for routing. We also design architectural connectivities which can exploit teleportation-based speedups. We then consider quantum routing by continuous-time Hamiltonian evolution, designing a routing algorithm that achieves a polynomial speedup in routing time over gate-based routing methods in architectures with a bottleneck, which is an intermediate region that constrains interactions between the rest of the device. We improve the lower bound on the time taken for routing in such systems, and develop a matching upper bound on the closely related task of generating and distributing entanglement. In the second part of this thesis, we investigate the behavior of finite-sized systems evolving by local Hamiltonians for a long duration. A major direction of research in condensed matter physics is the investigation of whether isolated systems thermalize (i.e., reach a state of thermal equilibrium). Various theoretical proposals such as information scrambling, entanglement between subsystems, and the eigenstate thermalization hypothesis have attempted to explain the conditions under which systems equilibrate. At the same time, notable counter-examples to thermalization have been found, such as systems that are integrable, exhibit many-body localization, or quantum many-body scars. We approach this question from a complexity-theoretic perspective. We show that the problem of deciding whether or not observables equilibrate to a givenvalue is PSPACE-complete. We also show that deciding whether observables relax to the thermal equilibrium value is contained in PSPACE, and is PSPACE-hard under quantum polynomial time reductions. These problems are thus intractable even for a quantum computer. Our results indicate that the proposed conditions leading to thermalization are either difficult to decide, or are not generic indicators of thermalization.
We present methods for implementing arbitrary permutations of qubits under interaction constraints. Our protocols make use of previous methods for rapidly reversing the order of qubits along a path. Given nearest-neighbor interactions on a path of length n , we show that there exists a constant ϵ≈0.034 such that the quantum routing time is at most (1−ϵ)n , whereas any swap-based protocol needs at least time n−1 . This represents the first known quantum advantage over swap-based routing methods and also gives improved quantum routing times for realistic architectures such as grids. Furthermore, we show that our algorithm approaches a quantum routing time of 2n/3 in expectation for uniformly random permutations, whereas swap-based protocols require time n asymptotically. Additionally, we consider sparse permutations that route k≤n qubits and give algorithms with quantum routing time at most n/3+O(k2) on paths and at most 2r/3+O(k2) on general graphs with radius r .
The Swap gate is a ubiquitous tool for moving information on quantum hardware, yet it can be considered a classical operation because it does not entangle product states. Genuinely quantum operations could outperform Swap for the task of permuting qubits within an architecture, which we call routing. We consider quantum routing in two models: (1) allowing arbitrary two-qubit unitaries, or (2) allowing Hamiltonians with norm-bounded interactions. We lower bound the circuit depth or time of quantum routing in terms of spectral properties of graphs representing the architecture interaction constraints, and give a generalized upper bound for all simple connected n-vertex graphs. In particular, we give conditions for a superpolynomial classical-quantum routing separation, which exclude graphs with a small spectral gap and graphs of bounded degree. Finally, we provide examples of a quadratic separation between gate-based and Hamiltonian routing models with a constant number of local ancillas per qubit and of an Ω(n) speedup if we also allow fast local interactions.
To implement arbitrary quantum circuits in architectures with restricted interactions, one may effectively simulate all-to-all connectivity by routing quantum information. We consider the entanglement dynamics and routing between two regions only connected through an intermediate “bottleneck” region with few qubits. In such systems, where the entanglement rate is restricted by a vertex boundary rather than an edge boundary of the underlying interaction graph, existing results such as the small incremental entangling theorem give only a trivial constant lower bound on the routing time (the minimum time to perform an arbitrary permutation). We significantly improve the lower bound on the routing time in systems with a vertex bottleneck. Specifically, for any system with two regions L , R with N L , N R qubits, respectively, coupled only through an intermediate region C with N C qubits, for any δ > 0 we show a lower bound of Ω ( N R 1 − δ / N L N C ) on the Hamiltonian quantum routing time when using piecewise time-independent Hamiltonians, or time-dependent Hamiltonians subject to a smoothness condition. We also prove an upper bound on the average amount of bipartite entanglement between L and C , R that can be generated in time t by such architecture-respecting Hamiltonians in systems constrained by vertex bottlenecks, improving the scaling in the system size from O ( N L t ) to O ( N L t ) . As a special case, when applied to the star graph (i.e., one vertex connected to N leaves), we obtain an Ω ( N 1 − δ ) lower bound on the routing time and on the time to prepare N / 2 Bell pairs between the vertices. We also show that, in systems of free particles, we can route optimally on the star graph in time Θ ( N ) using Hamiltonian quantum routing, obtaining a speedup over gate-based routing, which takes time Θ ( N ) .
Bidirectional quantum teleportation is a fundamental protocol for exchanging quantum information between two parties. Specifically, two individuals make use of a shared resource state as well as local operations and classical communication (LOCC) to swap quantum states. In this work, we concisely highlight the contributions of our companion paper [A. U. Siddiqui and M. M. Wilde, arXiv:2010.07905 (2020)]. We develop two different ways of quantifying the error of nonideal bidirectional teleportation by means of the normalized diamond distance and the channel infidelity. We then establish that the values given by both metrics are equal for this task. Additionally, by relaxing the set of operations allowed from LOCC to those that completely preserve the positivity of the partial transpose, we obtain semidefinite programing lower bounds on the error of nonideal bidirectional teleportation. We evaluate these bounds for some key examples—isotropic states and when there is no resource state at all. In both cases, we find an analytical solution. The second example establishes a benchmark for classical versus quantum bidirectional teleportation. Another example that we investigate consists of two Bell states that have been sent through a generalized amplitude damping channel. For this scenario, we find an analytical expression for the error, as well as a numerical solution that agrees with the former up to numerical precision.
Quantum networks play a pivotal role in quantum information science, which not only provide a secure communication platform for remote access to quantum computers but also serve as the strategic core for achieving large-scale quantum information processing, forming the foundational infrastructure for the future global-scale quantum internet. Quantum teleportation, which enables the transmission of unknown quantum states over long distances by employing quantum entanglement together with classical communication, is essential for the distribution of quantum resources in the construction of the global-scale quantum internet. To realize a global-scale quantum internet, quantum repeater protocols represent one of the most promising approaches for enabling quantum communication between any nodes. This concise review presents representative experimental demonstrations of quantum teleportation for constructing quantum networks across different physical platforms. Along this trajectory, the review discusses current challenges, open issues, and future perspectives toward scalable and practical quantum internet.