# Measuring Analytic Gradients of General Quantum Evolution with the Stochastic Parameter Shift Rule

@article{Banchi2020MeasuringAG, title={Measuring Analytic Gradients of General Quantum Evolution with the Stochastic Parameter Shift Rule}, author={Leonardo Banchi and Gavin E. Crooks}, journal={Quantum}, year={2020}, volume={5}, pages={386} }

Hybrid quantum-classical optimization algorithms represent one of the most promising application for near-term quantum computers. In these algorithms the goal is to optimize an observable quantity with respect to some classical parameters, using feedback from measurements performed on the quantum device. Here we study the problem of estimating the gradient of the function to be optimized directly from quantum measurements, generalizing and simplifying some approaches present in the literature…

## 63 Citations

### A variational toolbox for quantum multi-parameter estimation

- Computer Science, Physicsnpj Quantum Information
- 2021

This work demonstrates that variational quantum algorithms feasible on such devices address a challenge central to the field of quantum metrology: the identification of near-optimal probes and measurement operators for noisy multi-parameter estimation problems.

### Estimating the gradient and higher-order derivatives on quantum hardware

- Physics
- 2021

The authors show how to evaluate, with near-term quantum computers, high-order derivatives of expectation values with respect to the variational parameters of quantum circuits. The authors also study…

### A continuous variable Born machine

- Computer ScienceQuantum Mach. Intell.
- 2022

The continuous variable Born machine is discussed, built on the alternative architecture of continuous variable quantum computing, which is much more suitable for modelling such distributions in a resource-minimal way and can learn both quantum and classical continuous distributions, including in the presence of noise.

### A feasible approach for automatically differentiable unitary coupled-cluster on quantum computers†

- Computer ScienceChemical science
- 2021

It is shown that the gradient of an expectation value with respect to a parameterized n-fold fermionic excitation can be evaluated by four expectation values of similar form and size, whereas most standard approaches, based on the direct application of the parameter-shift-rule, come with an associated cost of expectation values.

### "Proper"Shift Rules for Derivatives of Perturbed-Parametric Quantum Evolutions

- Mathematics
- 2022

Banchi & Crooks ( Quantum, 2021; building 1951) have given methods to estimate derivatives of expectation values depending on a parameter that enters via a “perturbed” quantum evolution x 7→ e i ( xA…

### Iteration Complexity of Variational Quantum Algorithms

- Computer Science
- 2022

The iteration complexity of VQA is analyzed, that is, the number of steps V QA required until the iterates satisfy a surrogate measure of optimality, to derive the missing guarantees and show that the rate of convergence is unaﬀected.

### Optimizing Variational Quantum Algorithms with qBang: Efficiently Interweaving Metric and Momentum to Tackle Flat Energy Landscapes

- Computer Science
- 2023

The quantum Broyden adaptive natural gradient (qBang) approach is proposed, a novel optimizer that aims to distill the best aspects of existing approaches and introduces a new development strategy for gradient-based VQAs with a plethora of possible improvements.

### Here comes the $\mathrm{SU}(N)$: multivariate quantum gates and gradients

- Computer Science
- 2023

This work proposes a gate which fully parameterizes the special unitary group $\mathrm{SU}(N)$.

### Training variational quantum algorithms with random gate activation

- Computer Science
- 2023

A novel training algorithm with random quantum gate activation for VQAs to efficiently address barren plateau problem and it is proposed that the entanglement phase transition could be one underlying reason why the authors' RA training is so effective.

### Adiabatic quantum learning

- Physics
- 2023

Adiabatic quantum control protocols have been of wide interest to quantum computation due to their robustness and insensitivity to their actual duration of execution. As an extension of previous…

## 79 References

### Theory of variational quantum simulation

- PhysicsQuantum
- 2019

This work completes the theory of variational quantum simulation of general real and imaginary time evolution and it is applicable to near-term quantum hardware.

### Low-Depth Gradient Measurements Can Improve Convergence in Variational Hybrid Quantum-Classical Algorithms.

- Computer SciencePhysical review letters
- 2021

Within a natural black-box setting, a quantum variational algorithm that measures analytic gradients of the objective function with a low-depth circuit and performs stochastic gradient descent provably converges to an optimum faster than any algorithm that only measures the objectivefunction itself.

### Quantum circuit learning

- Computer Science, PhysicsPhysical Review A
- 2018

A classical-quantum hybrid algorithm for machine learning on near-term quantum processors, which is hybridizing a low-depth quantum circuit and a classical computer for machinelearning, paves the way toward applications of near- term quantum devices for quantum machine learning.

### Methodology for replacing indirect measurements with direct measurements

- Computer SciencePhysical Review Research
- 2019

These protocols can reduce the depth of the quantum circuit significantly by making the controlled operation unnecessary and hence are suitable for quantum-classical hybrid algorithms on near-term quantum computers.

### Evaluating analytic gradients on quantum hardware

- Computer SciencePhysical Review A
- 2019

This paper shows how gradients of expectation values of quantum measurements can be estimated using the same, or almost the same the architecture that executes the original circuit, and proposes recipes for the computation of gradients for continuous-variable circuits.

### Adam: A Method for Stochastic Optimization

- Computer ScienceICLR
- 2015

This work introduces Adam, an algorithm for first-order gradient-based optimization of stochastic objective functions, based on adaptive estimates of lower-order moments, and provides a regret bound on the convergence rate that is comparable to the best known results under the online convex optimization framework.

### npj Quantum Inf

- 2, 16019
- 2016

### Phys

- Rev. A 99, 032331
- 2019

### Phys

- Rev. A 98, 032309
- 2018

### Barren plateaus in quantum neural network training landscapes

- Computer Science, PhysicsNature Communications
- 2018

It is shown that for a wide class of reasonable parameterized quantum circuits, the probability that the gradient along any reasonable direction is non-zero to some fixed precision is exponentially small as a function of the number of qubits.