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Physical Review A

Effect of correlated decay on fault-tolerant quantum computation

B. Lemberger, Deniz D. Yavuz

Published Dec 29, 2017• 12 citations
Publisher

Abstract

We analyze noise in the circuit model of quantum computers when the qubits are coupled to a common bosonic bath and discuss the possible failure of scalability of quantum computation. Specifically, we investigate correlated (super-radiant) decay between the qubit energy levels from a two- or three-dimensional array of qubits without imposing any restrictions on the size of the sample. We first show that regardless of how the spacing between the qubits compares with the emission wavelength, correlated decay produces errors outside the applicability of the threshold theorem. This is because the sum of the norms of the two-body interaction Hamiltonians (which can be viewed as the upper bound on the single-qubit error) that decoheres each qubit scales with the total number of qubits and is unbounded. We then discuss two related results: (1) We show that the actual error (instead of the upper bound) on each qubit scales with the number of qubits. As a result, in the limit of large number of qubits in the computer, $N\ensuremath{\rightarrow}\ensuremath{\infty}$, correlated decay causes each qubit in the computer to decohere in ever shorter time scales. (2) We find the complete eigenvalue spectrum of the exchange Hamiltonian that causes correlated decay in the same limit. We show that the spread of the eigenvalue distribution grows faster with $N$ compared to the spectrum of the unperturbed system Hamiltonian. As a result, as $N\ensuremath{\rightarrow}\ensuremath{\infty}$, quantum evolution becomes completely dominated by the noise due to correlated decay. These results argue that scalable quantum computing may not be possible in the circuit model in a two- or three- dimensional geometry when the qubits are coupled to a common bosonic bath.

DOI
10.1103/physreva.96.062337

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We analyze noise in the circuit model of quantum computers when the qubits are coupled to a common bosonic bath and discuss the possible failure of scalability of quantum computation. Specifically, we investigate correlated (super-radiant) decay between the qubit energy levels from a two- or three-dimensional array of qubits without imposing any restrictions on the size of the sample. We first show that regardless of how the spacing between the qubits compares with the emission wavelength, correlated decay produces errors outside the applicability of the threshold theorem. This is because the sum of the norms of the two-body interaction Hamiltonians (which can be viewed as the upper bound on the single-qubit error) that decoheres each qubit scales with the total number of qubits and is unbounded. We then discuss two related results: (1) We show that the actual error (instead of the upper bound) on each qubit scales with the number of qubits. As a result, in the limit of large number of qubits in the computer, N\ensuremath→\ensuremath∞N\ensuremath{\rightarrow}\ensuremath{\infty}N\ensuremath→\ensuremath∞, correlated decay causes each qubit in the computer to decohere in ever shorter time scales. (2) We find the complete eigenvalue spectrum of the exchange Hamiltonian that causes correlated decay in the same limit. We show that the spread of the eigenvalue distribution grows faster with NNN compared to the spectrum of the unperturbed system Hamiltonian. As a result, as N\ensuremath→\ensuremath∞N\ensuremath{\rightarrow}\ensuremath{\infty}N\ensuremath→\ensuremath∞, quantum evolution becomes completely dominated by the noise due to correlated decay. These results argue that scalable quantum computing may not be possible in the circuit model in a two- or three- dimensional geometry when the qubits are coupled to a common bosonic bath.
We quantitatively analyze superradiance (collective emission) in a three-dimensional array of qubits without imposing any restrictions on the size of the sample. We show that even when the spacing between the qubits become arbitrarily large, superradiance produces an error rate on each qubit that scales with the total number of qubits. This is because the sum of the norms of the effective Hamiltonians that decoheres each qubit scales with the total number of qubits and is, therefore, unbounded. In three spatial dimensions, the sum of the norms scales as N2/3N^{2/3}N2/3 where NNN is the total number of qubits in the computer. Because the sum of the Hamiltonian norms are unbounded, the introduced errors are outside the applicability of the threshold theorem.
Over three decades after the formulation of early quantum algorithms, the realization of large-scale, fault-tolerant quantum computers remains constrained within the Noisy Intermediate-Scale Quantum (NISQ) regime. This paper critically re-evaluates the foundational assumptions underlying the Quantum Threshold Theorem and scalable Quantum Error Correction (QEC). We contrast classical error-suppression paradigms—specifically von Neumann’s majority-logic architecture for unreliable components—with the physical realities of continuous SU(2) symmetries in Hilbert space. We argue that the physical implementation of fault-tolerant quantum computing relies on a highly idealized model of uncorrelated, local Markovian noise, whose physical realizability at macroscopic scales remains contested. By examining interaction Hamiltonians in highly entangled macroscopic states, we demonstrate that collective decoherence channels and spatial noise correlations scale nonlinearly with register size n. This correlated phase drift acts as a directional force field—analogous to a long-wavelength hydrodynamic gradient across a field of coupled oscillators—rendering spatial averaging and independent syndrome measurements mathematically incapable of distinguishing systemic logic from collective environmental shifts. Consequently, we propose that universal quantum computers should not be conceptualized as scalable digital Turing machines, but rather as highly specialized, analog physical emulators bound by fundamental physical limits of continuous symmetry preservation.
We analyze the radiative coupling of qubits to a common radiation bath in free space and discuss that superradiance (collective emission) results in decoherence that scales with the square of the number of qubits. This decoherence mechanism is nonlocal; the collective emission simultaneously decoheres all the qubits in the computer. Because the noise source is not local, the introduced errors are outside the applicability of the threshold theorem. As a result, we argue that the errors due to superradiance cannot be corrected using the current models of fault-tolerant quantum computation.
We analyze the long-time behavior of a quantum computer running a quantum error correction (QEC) code in the presence of a correlated environment. Starting from a Hamiltonian formulation of realistic noise models, and assuming that QEC is indeed possible, we find formal expressions for the probability of a given syndrome history and the associated residual decoherence encoded in the reduced density matrix. Systems with nonzero gate times (``long gates'') are included in our analysis by using an upper bound on the noise. In order to introduce the local error probability for a qubit, we assume that propagation of signals through the environment is slower than the QEC period (hypercube assumption). This allows an explicit calculation in the case of a generalized spin-boson model and a quantum frustration model. The key result is a dimensional criterion: If the correlations decay sufficiently fast, the system evolves toward a stochastic error model for which the threshold theorem of fault-tolerant quantum computation has been proven. On the other hand, if the correlations decay slowly, the traditional proof of this threshold theorem does not hold. This dimensional criterion bears many similarities to criteria that occur in the theory of quantum phase transitions.
Abstract Quantum systems are inevitably open, continually exchanging energy and information with the surrounding environment. This interaction leads to decoherence and decay of quantum states. In complex systems, formed by many particles, decay can become correlated and enhanced. A fundamental question then arises: what is the maximal decay rate of a large quantum system, and how does it scale with its size? Computing this rate exactly is as hard as finding the ground-state energy of a generic spin Hamiltonian—a notoriously intractable problem. Here we exploit this correspondence to establish rigorous and general upper and lower bounds on the maximal decay rate. These bounds are universal, as they hold for a broad class of Markovian many-body quantum systems. For many physically relevant systems, the bounds are asymptotically tight, resulting in exact scaling laws with system size. Specifically, for large atomic arrays in free space, these scalings depend only on the dimensionality of the array and are insensitive to details at short length scales. The scaling laws set fundamental limits on the decay rates of all quantum states, shed light on the behaviour of generic driven-dissipative systems, and may ultimately constrain the scalability of quantum processors and simulators based on atomic arrays.
We investigate the efficacy of topological quantum error-correction in correlated noise model which permits collective coupling of all the codeword qubits to the same non-Markovian environment. In this noise model, the probability distribution over set of phase-flipped qubits, decays sub-exponentially in the size of the set and carries non-trivial likelihood of the occurring large numbers of qubits errors. We find that in the presence of noise correlation, one cannot guarantee arbitrary high computational accuracy simply by incrementing the codeword size while retaining constant noise level per qubit operation. However, if instead, per-operation qubit error probability in an n-qubits long codeword is reduced O(\sqrt{n}) times below the accuracy threshold, arbitrarily accurate quantum computation becomes feasible with acceptable scaling of the codeword size. Our results suggest that progressively reducing noise level in qubits and gates is as important as continuously integrating more qubits to realize scalable and reliable quantum computer.