Entanglement devised barren plateau mitigation

Open Access

Entanglement devised barren plateau mitigation

Taylor L. Patti, Khadijeh Najafi, Xun Gao, and Susanne F. Yelin

Phys. Rev. Research 3, 033090 – Published 23 July 2021

Abstract

Hybrid quantum-classical variational algorithms are one of the most propitious implementations of quantum computing on near-term devices, offering classical machine-learning support to quantum scale solution spaces. However, numerous studies have demonstrated that the rate at which this space grows in qubit number could preclude learning in deep quantum circuits, a phenomenon known as barren plateaus. In this work, we implicate random entanglement, i.e., entanglement that is formed due to state evolution with random unitaries, as a source of barren plateaus and characterize them in terms of many-body entanglement dynamics, detailing their formation as a function of system size, circuit depth, and circuit connectivity. Using this comprehension of entanglement, we propose and demonstrate a number of barren plateau ameliorating techniques, including initial partitioning of cost function and non-cost function registers, meta-learning of low-entanglement circuit initializations, selective inter-register interaction, entanglement regularization, the addition of Langevin noise, and rotation into preferred cost function eigenbases. We find that entanglement limiting, both automatic and engineered, is a hallmark of high-accuracy training and emphasize that, because learning is an iterative organization process whereas barren plateaus are a consequence of randomization, they are not necessarily unavoidable or inescapable. Our work forms both a theoretical characterization and a practical toolbox; first defining barren plateaus in terms of random entanglement and then employing this expertise to strategically combat them.

Received 22 December 2020
Accepted 21 June 2021

DOI:https://doi.org/10.1103/PhysRevResearch.3.033090

Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article's title, journal citation, and DOI.

Published by the American Physical Society

Physics Subject Headings (PhySH)

Machine learning Quantum algorithms Quantum computation Quantum entanglement

Quantum Information, Science & Technology

Authors & Affiliations

Taylor L. Patti ^1,*, Khadijeh Najafi², Xun Gao¹, and Susanne F. Yelin¹

¹Department of Physics, Harvard University, Cambridge, Massachusetts 02138, USA
²Department of Physics, Harvard University, Cambridge, Massachusetts 02138, USA; IBM Quantum, IBM T.J. Watson Research Center, Yorktown Heights, New York 10598 USA; and Division of Physics, Mathematics and Astronomy, California Institute of Technology, Pasadena, California 91125, USA

^*taylorpatti@g.harvard.edu

Article Text

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Issue

Vol. 3, Iss. 3 — July - September 2021

Subject Areas

Quantum Physics

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Images

Figure 1
(a) Diagram of a linear circuit with $n_{C} = 3$ , $n_{N} = 2$ , and $L = 4$ . Qubits $q_{i}$ and $q_{j}$ interact in layers $k$ through two-qubit unitaries $u_{i j}^{k}$ , which comprise layer unitaries $U_{k}$ , and ultimately form total unitary $U$ . The qubits of $R_{C}$ are then read out by the cost function operator $M_{C}$ . (b) Ground-state compressor for randomly generated 9-qubit long-range interaction Hamiltonian [Eq. (8)] ground states. The circuit learns to represent the ground states $| Ψ_{g} 〉$ as 3-qubit representations $| ψ_{g} 〉$ . (c) An illustration of barren plateaus with $L = 〈 σ_{1}^{z} σ_{2}^{z} 〉$ ( $n_{C} = 2$ ) and $n = 3, 5, 7, 9$ (blue, orange, green, red), with each data point sampling two-thousand distinct circuit iterations. The variance $σ_{O}^{2}$ of the partial cost function derivative $O$ is known to decrease rapidly with increasing circuit layers $L$ until ultimately reaching the plateau magnitude $σ_{B}^{2} \propto 2^{- n}$ .
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Figure 2
(a) Variance $σ_{O}^{2}$ of cost function derivative $O$ for $L = 〈 \prod_{i = 1}^{n_{C}} σ_{i}^{z} 〉$ in units of its barren plateau value $σ_{B}^{2}$ [horizontal asymptote for each $n$ in Fig. 1] vs number of gate layers, $L$ , for $n = 9$ and various $n_{C}$ . The inset shows the change in entanglement entropy $S$ vs circuit depth $L$ for a 2–3 qubit bipartition, as defined in Eq. (11) and illustrated in the inset of panel (b). As the $n_{C} = 4$ system has an entropy bipartition which is exactly aligned with the $R_{C} − R_{N}$ partitioning, $S$ describes the entanglement growth analytically for this case, while it is only an approximation for $n_{C} = 1$ , 2, 3. This approximation could be improved by using an initial $n_{C} − n_{N}$ bipartition. (b) Variance $σ_{O}^{2}$ of cost function derivative $O$ in units of its barren plateau value $σ_{B}^{2}$ vs normalized change in entropy $S$ for $L = 〈 σ_{1}^{z} σ_{2}^{z} 〉$ ( $n_{C} = 2$ ) and $n = 3, 5, 7, 9$ (blue, orange, green, and red). Larger values of $n$ experience greater relative suppression of $σ_{O}^{2}$ as $σ_{O}^{2} / σ_{B}^{2} \propto S_{B} = 2^{- S}$ , which is the analytical solution for $n = 5$ (orange) and an approximation for other $n$ . The inset shows a schematic of bipartition entropy of entanglement $S$ for the smaller system $n = 5$ . Full density matrix $ρ$ broken into two subsets $R_{α}$ and $R_{β}$ , where $R_{α}$ always contains as much of $R_{C}$ as possible. In all subfigures, each data point sampling two-thousand distinct circuit iterations.
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Figure 3
(a) Partitioned cost function derivative variance $σ_{OP}^{2}$ for $L = 〈 σ_{1}^{z} σ_{2}^{z} 〉$ in units of its nonpartitioned value $σ_{O}^{2}$ vs number of gate layers $L$ for $n = 3, 5, 7, 9$ (blue, orange, green, and red) and $n_{C} = 2$ . As expected, $σ_{OP}^{2}$ is approximately a factor of $2^{n_{N}}$ larger than $σ_{O}^{2}$ due to the absence of random $R_{C} − R_{N}$ entanglement. (b) Training loss $L = L_{g}$ of Eq. (9) for ground-state compressor in Fig. 1 (gray) and $L = | 〈 σ_{1}^{z} σ_{2}^{z} σ_{3}^{z} 〉 |$ (red) vs training epochs for $L = 200$ . Solid lines are initially partitioned circuits and dashed lines are fully random initializations, with increased learning performance in the latter. The corresponding evolution of $S$ is shown in the inset. In both subfigures, each data point sampling two-thousand distinct circuit iterations.
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Figure 4
(a) A deep circuit ( $L = 100$ ) pretraining procedure for $L = 〈 σ_{1}^{z} σ_{2}^{z} 〉$ that minimizes collective entanglement $S_{C}$ [Eq. (21)], the entanglement entropy between both the input and output registers of $R_{C}$ and $R_{N}$ for $n = 3$ , 5 (blue, orange). As random entanglement decreases with $S_{C}$ , $σ_{O}^{2}$ increases. Crucially, this pretraining procedure is unique from partitioned initialization as it permits nontrivial interaction at the level of individual circuit layers (inset), with magnitude of inter-register interaction remaining $2 / π \approx 0.637$ , which is consistent with that of random $θ_{i}$ . (b) Derivative variance $σ_{O}^{2}$ for $L = 〈 σ_{1}^{z} σ_{2}^{z} 〉$ ( $n_{C} = 2$ ) and initially partitioned registers $R_{C}$ and $R_{N}$ in units of its nonpartitioned value $σ_{B}^{2}$ vs number of register entangling gate layers $L_{E}$ . Here, $L = 200$ and $n = 3, 5, 7, 9$ (blue, orange, green, and red). In both subfigures, each data point sampling two-thousand distinct circuit iterations.
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Figure 5
Loss function vs epochs averaged over two-thousand iterations of both entanglement regularized and initially partitioned circuits (solid lines) and initially partitioned only circuits (dashed lines) of (a) ground-state compressor $L = L_{g}$ [Eq. (9)], (b) $L = | 〈 σ_{1}^{z} σ_{2}^{z} σ_{3}^{z} 〉 |$ , and (c) $L = 〈 σ_{1}^{z} σ_{2}^{z} σ_{3}^{z} 〉$ with $L = 200$ . In all cases, nonzero regularization terms lead to (c) equal or (a), (b) faster and more accurate learning with decreased $S$ , mitigating the effects of barren plateaus. Moreover, the behavior of $S$ is reflective of the overall training difficulty (insets), whereas the cost functions of panels (a) and (c) can be learned rather rapidly and accurately and do so with naturally lower levels of bipartite entanglement $S$ that are more responsive to regularization, that of panel (b) trains slower and with less accuracy, with entanglement growth that is controlled very little by regularization. The black dashed line in panel (c) corresponds to $S$ for unregularized, unpartitioned $L = 〈 σ_{1}^{z} σ_{2}^{z} σ_{3}^{z} 〉$ , which learns with equal effectiveness as its partitioned and regularized counterparts, highlighting the increased learnability eigenstate learning.
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Figure 6
Preparing a state $| ψ 〉$ under barren plateau conditions (random initialization of $L = 200$ ) for $n = 9$ and cost function $L = | 〈 σ_{1}^{z} σ_{2}^{z} σ_{3}^{z} 〉 |$ through both (a) the addition of Langevin noise on a subset of parameters and (b) substitution of measurement basis for which the target state is an eigenstate. (a) Additional Langevin noise term $λ \sum_{i}^{N} | ϕ_{i} | L$ increases the gradient variance $σ_{O_{i}}^{2} \to {(1 + λ N π)}^{2} σ_{O_{i}}^{2}$ with respect to parameters $ϕ_{i}$ , thus helping to navigate barren plateau landscapes. This can be viewed as an increase in diffusion constant $2 D$ . (b) By substituting the cost function $〈 σ_{1}^{z} σ_{2}^{z} σ_{3}^{x} 〉$ , for which our target state $| ψ 〉$ is an eigenstate (or alternatively, choosing a “natural” cost function basis), we obtain a faster, more accurate learning process. Both experiments are the average over two-thousand distinct randomly parametrized circuits.
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