In a new paper, researchers from Rice University, IBM and the University of Southern California propose a novel quantum state reconstruction algorithm that extends the applicability of quantum tomography to larger systems and is orders of magnitude faster than state-of-the-art approaches while matching or improving accuracy.
Quantum tomography is a quantum state reconstruction process that uses measurements on an ensemble of identical quantum states. It is typically used for identifying the nature of imperfections and deviations in quantum processing unit (QPU) implementation. Quantum tomography can be applied for example to measure the signal gain or loss of optical devices, and in quantum computing and quantum information theory to reliably determine the actual states of qubits.
In general, quantum tomography is a non-scalable protocol. Quantum state tomography (QST) suffers from two main bottlenecks: the large amount of data required to perform tomography, and the numerical searching in an exponentially large space for a density matrix that is consistent with the data.
To improve the scalability of QST and address its computational bottleneck, the researchers combined ideas from compressed sensing, non-convex optimization and acceleration techniques to introduce a new algorithm — Momentum Inspired Factored Gradient Descent (MiFGD) — that pushes QST beyond its current capabilities.
The research team summarizes their contributions as:
- Prove that the non-convex MiFGD algorithm has indeed a favourable linear convergence rate, in terms of iterate distance, in the noiseless measurement data case and under common assumptions.
- Provide QST results based on real data from IBM’s quantum computers up to 8-qubits, contributing to recent efforts on testing QST algorithms in real quantum data. Their synthetic examples scale up to 12-qubits effortlessly, leaving the space for an efficient, hardware-aware implementation open for future work.
- Show in practice that MiFGD allows faster estimation of quantum states compared to state-of-the-art convex and non-convex algorithms, including recent deep learning approaches, even in the presence of statistical noise in the measurement data.
- Exploit parallel computations in MiFGD by extending its implementation to enable efficient, parallel execution over shared and distributed memory systems. In this way, they experimentally showcase the scalability of the proposed approach, which is particularly critical for tackling larger quantum system sizes.
- Provide an implementation of the approach, compatible with the open-source software Qiskit.
The MiFGD algorithm focuses on the factorized form of low-rank QST problems. The researchers first make certain assumptions on the problem parameters, including that the linear map satisfies the restricted isometry property; and introduce their central theorem: that MiFGD converges linearly to a neighbourhood of the optimal solution while using acceleration motions in a non-convex setting.
The MiFGD algorithm design is informed by three ideas from previous work in this area: compressed sensing, non-convex optimization, and acceleration techniques. Basically, compressed sensing QST is based on convex optimization methods used to obtain an estimation of the low-rank state. There are however two drawbacks to convex optimization: the search space grows exponentially with the number of qubits, and the optimization requires projection onto the PSD cone (set of all positive semidefinite matrices) at every iteration, which becomes exponentially more difficult with the number of qubits.
In their study, the researchers cleverly avoid these drawbacks by working in a factorized space. This use of factorization results in a search space that is substantially smaller than its convex counterpart and requires only a single use of top-r SVD (singular value decomposition) during the entire execution algorithm. Taken together with accelerating motions, the approach is able to estimate quantum states of qubit systems larger than those handled by state of the art methods.
In the next step, the team uses both simulated and real data to demonstrate that MiFGD outperforms non-accelerated methods. The results are for 6- and 8-qubit real data on the 20-qubit IBM Quantum Processing Unit “ibmq_boeblingen.”
The team compared MiFGD with three neural-network approaches, Positive Real Wave Function (PRWF), Complex Wave Function (CWF), and Density Matrix (DM), with MiFGD achieving high fidelity on all three states, GHZ(n), Hadamard(n) and Random(n). In most cases, MiFGD obtained high-fidelity solutions within seconds. For instance, on Random(8), MiFGD achieved 0.939 fidelity within 22.81 seconds, while PRWF required 2196.26 seconds to obtain 0.05 fidelity. The CWF and DM methods did not complete a single iteration within the three-hour time limit.
The team has provided a publicly available implementation of their approach, which is compatible with the open-source software Qiskit. They’ve also advanced MiFGD by extending its implementation to enable efficient, parallel execution over shared and distributed memory systems.
The paper Fast Quantum State Reconstruction via Accelerated Non-Convex Programming is on arXiv.
Author: Hecate He | Editor: Michael Sarazen
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