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arXiv:2303.10860 · npj Quantum Information

Probabilistic state synthesis based on optimal convex approximation

Seiseki Akibue, Go Kato, Seiichiro Tani

Published Jan 3, 2024• 5 citations
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Abstract

Abstract When preparing a pure state with a quantum circuit, there is an unavoidable approximation error due to the compilation error in fault-tolerant implementation. A recently proposed approach called probabilistic state synthesis, where the circuit is probabilistically sampled, is able to reduce the approximation error compared to conventional deterministic synthesis. In this paper, we demonstrate that the optimal probabilistic synthesis quadratically reduces the approximation error. Moreover, we show that a deterministic synthesis algorithm can be efficiently converted into a probabilistic one that achieves this quadratic error reduction. We also numerically demonstrate how this conversion reduces the T -count and analytically prove that this conversion halves an information-theoretic lower bound on the circuit size. In order to derive these results, we prove general theorems about the optimal convex approximation of a quantum state. Furthermore, we demonstrate that this theorem can be used to analyze an entanglement measure.

DOI
10.1038/s41534-023-00793-7

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Abstract When preparing a pure state with a quantum circuit, there is an unavoidable approximation error due to the compilation error in fault-tolerant implementation. A recently proposed approach called probabilistic state synthesis, where the circuit is probabilistically sampled, is able to reduce the approximation error compared to conventional deterministic synthesis. In this paper, we demonstrate that the optimal probabilistic synthesis quadratically reduces the approximation error. Moreover, we show that a deterministic synthesis algorithm can be efficiently converted into a probabilistic one that achieves this quadratic error reduction. We also numerically demonstrate how this conversion reduces the T -count and analytically prove that this conversion halves an information-theoretic lower bound on the circuit size. In order to derive these results, we prove general theorems about the optimal convex approximation of a quantum state. Furthermore, we demonstrate that this theorem can be used to analyze an entanglement measure.
Pure quantum states are often approximately encoded as classical bit strings such as those representing probability amplitudes and those describing circuits that generate the quantum states. The crucial quantity is the minimum length of classical bit strings from which the original pure states are approximately reconstructible. We derive asymptotically tight bounds on the minimum bit length required for probabilistic encodings with which one can approximately reconstruct the original pure state as an ensemble of the quantum states encoded in classical strings. We also show that such a probabilistic encoding asymptotically halves the bit length required for "deterministic" ones. This is based on the fact that the accuracy of approximating pure states by using a given subset of pure states can be increased quadratically if we use ensembles of pure states in the subset. Moreover, we show that a similar fact holds when we consider the approximation of unitary gates by using a given subset of unitary gates. This improves the reduction rate of the circuit size by using probabilistic circuit synthesis compared to previous results. This also demonstrates that the reduction is possible even for low-accuracy circuit synthesis, which might improve the accuracy of various NISQ algorithms.
The purpose of unitary synthesis is to find a gate sequence that optimally approximates a target unitary transformation. A new synthesis approach, called probabilistic synthesis, has been introduced, and its superiority has been demonstrated over traditional deterministic approaches with respect to approximation error and gate length. However, the optimality of current probabilistic synthesis algorithms is unknown. We obtain the tight lower bound on the approximation error obtained by the optimal probabilistic synthesis, which guarantees the sub-optimality of current algorithms. We also show its tight upper bound, which improves and unifies current upper bounds depending on the class of target unitaries. These two bounds reveal the fundamental relationship of approximation error between probabilistic approximation and deterministic approximation of unitary transformations. From a computational point of view, we show that the optimal probability distribution can be computed by the semidefinite program (SDP) we construct. We also construct an efficient probabilistic synthesis algorithm for single-qubit unitaries, rigorously estimate its time complexity, and show that it reduces the approximation error quadratically compared with deterministic algorithms.
We develop error-tolerant quantum state discrimination(QSD) strategies that maintain reliable performance under moderate noise. Two complementary approaches are proposed: CrossQSD, which generalizes unambiguous discrimination with tunable confidence bounds to balance accuracy and efficiency, and FitQSD, which optimizes the measurement outcome distribution to approximate that of the ideal noiseless case. Furthermore, we provide a unified hybrid-objective QSD framework that continuously interpolates between minimum-error discrimination (MED) and FitQSD, allowing flexible trade-offs among competing objectives. The associated optimization problems are formulated as convex programs and efficiently solved via disciplined convex programming or, in many cases, semidefinite programming. Additionally, a circuit synthesis framework based on a modified Naimark dilation and isometry synthesis enables hardware-efficient implementations with substantially reduced qubit and gate resources. An open-source toolkit automates the full optimization and synthesis workflow, providing a practical route to QSD on current quantum devices.
To ensure resilience against the unavoidable noise in quantum computers, quantum information needs to be encoded using an error-correcting code, and circuits must have a particular structure to be fault-tolerant. Compilation of fault-tolerant quantum circuits is thus inherently different from the non-fault-tolerant case. However, automated fault-tolerant compilation methods are widely underexplored, and most known constructions are obtained manually for specific codes only. In this work, we focus on the problem of automatically synthesizing fault-tolerant circuits for the deterministic initialization of an encoded state for a broad class of quantum codes that are realizable on current and near-term hardware. To this end, we utilize methods based on techniques from classical circuit design, such as satisfiability solving, resulting in tools for the synthesis of (optimal) fault-tolerant state preparation circuits for near-term quantum codes. We demonstrate the correct fault-tolerant behavior of the synthesized circuits using circuit-level noise simulations. We provide all routines as open-source software as part of the Munich Quantum Toolkit (MQT) at https://github.com/cdatum/mqt-qecc.
Successful implementations of quantum technologies require protocols and algorithms that use as few quantum resources as possible. However, many important quantum operations, such as continuous rotation gates in quantum computing or broadband pulses in NMR or magnetic resonance imaging (MRI) applications, can only be implemented approximately using finite quantum resources. This work develops an approach that, on average—at the cost of a modestly increased measurement repetition rate—enables exact implementations. One proceeds by first building a library of a large number of different approximations to the desired gate operation; by randomly selecting these operations according to a preoptimized probability distribution, one can on average implement the desired operation with a rigorously controllable approximation error. The approach relies on sophisticated tools from convex optimization to efficiently find optimal probability distributions. A diverse spectrum of applications are demonstrated as (a) exactly synthesizing rotations in fault-tolerant quantum computers using only low T -count circuits and (b) synthesizing broadband and band-selective pulses of superior performance in quantum optimal control with (c) further applications in NMR or MRI. The approach is very general and a broad spectrum of practical applications in quantum technologies are explicitly demonstrated. Published by the American Physical Society 2024
Quantum Circuit Pre-Synthesis: Learning Local Edits to Reduce TTT-count
Compiling quantum circuits into Clifford+TTT gates is a central task for fault-tolerant quantum computing using stabilizer codes. In the near term, TTT gates will dominate the cost of fault tolerant implementations, and any reduction in the number of such expensive gates could mean the difference between being able to run a circuit or not. While exact synthesis is exponentially hard in the number of qubits, local synthesis approaches are commonly used to compile large circuits by decomposing them into substructures. However, composing local methods leads to suboptimal compilations in key metrics such as TTT-count or circuit depth, and their performance strongly depends on circuit representation. In this work, we address this challenge by proposing \textsc{Q-PreSyn}, a strategy that, given a set of local edits preserving circuit equivalence, uses a RL agent to identify effective sequences of such actions and thereby obtain circuit representations that yield a reduced TTT-count upon synthesis. Experimental results of our proposed strategy, applied on top of well-known synthesis algorithms, show up to a 20%20\%20% reduction in TTT-count on circuits with up to 25 qubits, without introducing any additional approximation error prior to synthesis.