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arXiv:2008.10611 · Quantum

How Dynamical Quantum Memories Forget

Lukasz Fidkowski, Jeongwan Haah, Matthew B. Hastings

Published Jan 17, 2021• 100 citations
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Abstract

Motivated by recent work showing that a quantum error correcting code can be generated by hybrid dynamics of unitaries and measurements, we study the long time behavior of such systems. We demonstrate that even in the ``mixed'' phase, a maximally mixed initial density matrix is purified on a time scale equal to the Hilbert space dimension (i.e., exponential in system size), albeit with noisy dynamics at intermediate times which we connect to Dyson Brownian motion. In contrast, we show that free fermion systems — i.e., ones where the unitaries are generated by quadratic Hamiltonians and the measurements are of fermion bilinears — purify in a time quadratic in the system size. In particular, a volume law phase for the entanglement entropy cannot be sustained in a free fermion system.

DOI
10.22331/q-2021-01-17-382

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Motivated by recent work showing that a quantum error correcting code can be generated by hybrid dynamics of unitaries and measurements, we study the long time behavior of such systems. We demonstrate that even in the ``mixed'' phase, a maximally mixed initial density matrix is purified on a time scale equal to the Hilbert space dimension (i.e., exponential in system size), albeit with noisy dynamics at intermediate times which we connect to Dyson Brownian motion. In contrast, we show that free fermion systems — i.e., ones where the unitaries are generated by quadratic Hamiltonians and the measurements are of fermion bilinears — purify in a time quadratic in the system size. In particular, a volume law phase for the entanglement entropy cannot be sustained in a free fermion system.
We introduce a continuous-time model of many-body quantum dynamics based on infinitesimal random unitary operations, combined with projective measurements. We consider purification dynamics in this model, where the system is initialized in a mixed state, which then purifies over time as a result of the measurements. By mapping our model to a family of effective 1D quantum Hamiltonians, we are able to derive analytic expressions that capture how the entropy of the system decays in time. Our results confirm the existence of two distinct dynamical phases, where purification occurs over a timescale that is exponential versus constant in system size. We compare our analytic expressions for this microscopic model to results derived from field theories that are expected to capture such measurement-induced phase transitions and find quantitative agreement between the two.
Fresh qubits, not size, set how long a noisy quantum system can remember. With kkk input qubits reinitialized per time step, the longest memory any reservoir can attain under depolarizing noise is exp⁡[Θ(k)]\exp[Θ(k)]exp[Θ(k)] time steps, whatever its size, dynamics, and input encoding. A universal upper bound and an explicit reservoir that attains the exponential scaling establish this optimal resource law. The bound depends on the dynamics only through the entropy that fresh qubits remove per time step, so it holds for any Markovian noisy dynamics without feedback, including measurement-free error correction.
We study purification dynamics in monitored quantum processes governed by ensembles of quantum circuits in different random-matrix symmetry classes. We analyze the universal aspects that emerge away from the measurement induced phase transition and inside the volume/weak measurement phase and in the scaling limit of large time and Hilbert space dimension. We present two toy models that reveal two complementary visions and provide quantitative access to universal scaling: i) a discrete-time dynamic in which each time step corresponds to multiplication by a Gaussian random matrix; ii) weak continuous-time monitoring that induces a Dyson brownian motion of the eigenvalues of the density matrix. The first approach provides an algebraic characterization based on rotational invariance emerging in Kraus's operator space, focusing in particular on the unitary and orthogonal cases, respectively β=2β=2β=2 and β=1β=1β=1, with βββ the Dyson random-matrix index. The second approach, on the other hand, allows for a unified treatment for any βββ, thanks to the mapping of the Fokker-Planck evolution of eigenvalues onto the Calogero-Sutherland integrable Hamiltonian diagonalized in terms of Jack polynomials. We provide explicit expressions for the universal decrease of Rényi entropies. We show that, approaching the universal scaling limit, numerical simulations of different models agree with each other and with our theoretical predictions. Our results clarify the existence of different classes of universality for the purification process in hybrid quantum systems, accessible in random circuit architectures and weak measurement protocols.
We consider a toy model for the study of monitored dynamics in many-body quantum systems. We study the stochastic Schrödinger equation resulting from continuous monitoring with a rate Γ of a random Hermitian operator, drawn from the Gaussian unitary ensemble (GUE) at every time t. Due to invariance by unitary transformations, the dynamics of the eigenvalues {λα}α=1n of the density matrix decouples from that of the eigenvectors, and is exactly described by stochastic equations that we derive. We consider two regimes: in the presence of an extra dephasing term, which can be generated by imperfect quantum measurements, the density matrix has a stationary distribution, and we show that in the limit of large size n→∞ it matches with the inverse-Marchenko–Pastur distribution. In the case of perfect measurements, instead, purification eventually occurs and we focus on finite-time dynamics. In this case, remarkably, we find an exact solution for the joint probability distribution of λ’s at each time t and for each size n. Two relevant regimes emerge: at short times tΓ=O(1), the spectrum is in a Coulomb gas regime, with a well-defined continuous spectral distribution in the n→∞ limit. In that case, all moments of the density matrix become self-averaging and it is possible to exactly characterize the entanglement spectrum. In the limit of large times tΓ=O(n), one enters instead a regime in which the eigenvalues are exponentially separated log(λα/λβ)=O(Γt/n), but fluctuations ∼O(Γt/n) play an essential role. We are still able to characterize the asymptotic behaviors of the entanglement entropy in this regime.
We study entanglement fluctuations and quantum error correction in the weakly monitored volume-law phase of quantum automaton circuits subject to repeated local measurements. We numerically observe that the entanglement entropy exhibits strong fluctuation with the exponent close to the ``growth exponent'' of the Kardar-Parisi-Zhang (KPZ) universality class, the same as other local random circuits studied previously. We also investigate the dynamically generated quantum error correction code in the purification process and show that this model has different contiguous code distances for two types of errors that exhibit similar sublinear power-law scaling. We give an interpretation of these results by mapping them to various quantities in a classical particle model. We demonstrate that the subleading correction term of the entanglement entropy and the sublinear power-law scaling of the contiguous code distance in the volume-law phase are both the emergent phenomena of the hybrid random dynamics. Finally, we show that this classical particle dynamics itself has a type of error correction ability and can dynamically generate a classical linear code.
Fermionic platforms offer compelling architectures for quantum computing, ranging from topologically protected Majorana-based qubits to fermionic cold atoms. To achieve scalability, however, they require quantum error correction. In this work, we prove that any exact and sufficiently accurate approximate fermionic quantum error correction necessarily requires non-Gaussian operations, beyond the free-fermion regime of quadratic dynamics. This is in sharp contrast to the qubit setting, where the efficiently classically simulable stabilizer operations form the standard framework for quantum error correction. Specifically, we show that the logical space of any non-trivial fermionic error-correcting code contains no pure fermionic Gaussian state, utilizing a fundamental incompatibility between fermionic error correction and Wick's theorem. We further show that the required number of bounded-weight non-Gaussian gates for unitary codeword preparation grows at least linearly with both the code distance and the number of encoded modes, revealing an intrinsic resource overhead that increases simultaneously with error-protection strength and logical capacity. Furthermore, we analyze the performance of fermionic Gaussian operations in entanglement distillation, revealing a distinction from their bosonic counterparts. Our results reveal fundamental difficulties for fermionic error correction from the perspectives of both physical implementation and classical simulation, suggesting connections to fermionic phases of matter and state preparation complexity.