Quantum Information

For quantum phases of Hamiltonian ground states, the energy gap plays a central role in ensuring the stability of the phase as long as the gap remains finite. In this talk we introduce Markov length, the length scale at which the quantum conditional mutual information (CMI) decays exponentially, as an equally essential quantity characterizing mixed-state phases and transitions. For a state evolving under a local Lindbladian, we argue that if its Markov length remains finite along the evolution, then it remains in the same phase, meaning there exists another quasi-local Lindbladian evolution that can reverse the former one. We apply this diagnostic to toric code subject to decoherence and show that the Markov length is finite everywhere except at its decodability transition, at which it diverges. This implies that the mixed state phase transition coincides with the decodability transition and also suggests a quasi-local decoding channel.

Ground states as well as Gibbs states of many-body quantum Hamiltonians have been studied extensively for some time. In contrast, the landscape of mixed states that do not correspond to a system in thermal equilibrium is relatively less explored. In this talk I will motivate a rather coarse characterization of mixed quantum many-body states using the idea of "separability", i.e., whether a mixed state can be expressed as an ensemble of short-range entangled pure states. I will discuss several examples of decoherence-driven phase transitions from a separability viewpoint, and argue that such a framework also provides a potentially new view on Gibbs states. Based on work with Yu-Hsueh Chen. References: 2309.11879, 2310.07286, 2403.06553.

"Optimal fault tolerant error correction thresholds for CCS codes are traditionally obtained via mappings to classical statistical mechanics models, for example the 2d random bond Ising model for the 1d repetition code subject to bit-flip noise and faulty measurements. Here, we revisit the 1d repetition code, and develop an exact “stabilizer expansion” of the full time evolving density matrix under repeated rounds of (incoherent and coherent) noise and faulty stabilizer measurements. This expansion enables computation of the coherent information, indicating whether encoded information is retained under the noisy dynamics, and generates a dual representation of the (replicated) 2d random bond Ising model. However, in the fully generic case with both coherent noise and weak measurements, the stabilizer expansion breaks down (as does the canonical 2d random bond Ising model mapping). If the measurement results are thrown away all encoded information is lost at long times, but the evolution towards the trivial steady state reveals a signature of a quantum transition between an over and under damped regime. Implications for generic noisy dynamics in other CCS codes will be mentioned, including open issues."

We define a map from an arbitrary quantum circuit to a local Hamiltonian whose ground state encodes the quantum computation. All previous maps relied on the Feynman-Kitaev construction, which introduces an ancillary ‘clock register’ to track the computational steps. Our construction, on the other hand, relies on injective tensor networks with associated parent Hamiltonians, avoiding the introduction of a clock register. This comes at the cost of the ground state containing only a noisy version of the quantum computation, with independent stochastic noise. We can remedy this - making our construction robust - by using quantum fault tolerance. In addition to the stochastic noise, we show that any state with energy density exponentially small in the circuit depth encodes a noisy version of the quantum computation with adversarial noise. We also show that any ‘combinatorial state’ with energy density polynomially small in depth encodes the quantum computation with adversarial noise. This serves as evidence that any state with energy density polynomially small in depth has a similar property. As an application, we give a new proof of the QMA-completeness of the local Hamiltonian problem (with logarithmic locality) and show that contracting injective tensor networks to additive error is BQP- hard. We also discuss the implication of our construction to the quantum PCP conjecture, combining with an observation that QMA verification can be done in logarithmic depth. Based on joint work with Anurag Anshu and Nikolas P. Breuckmann. (https://arxiv.org/abs/2309.16475)

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Quantum measurement has conventionally been regarded as the final step in quantum information processing, which is essential for reading out the processed information but collapses the quantum state into a classical state. However, recent studies have shown that quantum measurement itself can induce novel quantum phenomena. One seminal example is a monitored random circuit, which can generate long-range entanglement faster than a random unitary circuit. Inspired by these results, in this talk, we address the following question: When quantum information is encoded in a quantum error-correcting code, how many physical qubits should be randomly measured to destroy the encoded information? We investigate this question for various quantum error-correcting codes and derive the necessary and sufficient conditions for destroying the information through measurements. In particular, we demonstrate that for a large class of quantum error-correcting codes, it is impossible to destroy the encoded information through random single-qubit Pauli measurements when a tiny portion of physical qubits is still unmeasured. Our results not only reveal the extraordinary robustness of quantum codes under measurement decoherence, but also suggest potential applications in quantum information processing tasks.

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In the one-way model of measurement based quantum computing, unlike the quantum circuit model, a computation is driven not by unitary gates but by successive adaptive single-qubit measurements on an entangled resource state. So-called flow properties ensure that a one-way computation, described by a measurement pattern, is deterministic overall (up to Pauli corrections on output qubits). Translations between quantum circuits and measurement patterns have been used to show universality of the one-way model, verify measurement patterns, optimise quantum circuits, and more. Yet while it is straightforward to translate a circuit into a measurement pattern, the question of algorithmic "circuit extraction" -- how to translate general measurement patterns with flow to ancilla-free circuits -- had long remained open for all but the simplest type of flow. In this talk, we will recap the one-way model of quantum computing and then explain how the problem of circuit extraction was resolved using the ZX-calculus as a common language for circuits and measurement patterns. We also discuss applications.