# Google Quantum AI¶

Today I wanted to get an insight into Google’s research activities, and I reviewed the publicly available research publications since 2017 as listed on Quantum AI. I focussed on three areas: quantum advantage, quantum chemistry, quantum computation (linear system solvers, QAOA, QML). - December 12, 2022 (updated Apr 30, 2023)

## Quantum Advantage¶

- Mohseni et al.,
**Commercialize Quantum Technologies in Five Years**, 2017, research.google:45919. - Boixo et al.,
**Characterizing Quantum Supremacy in Near-Term Devices**, 2018, research.google:46227. - Markov et al.,
**Quantum Supremacy Is Both Closer and Farther than It Appears**, 2018, research.google:47252. - Arute et al.,
**Quantum Supremacy using a Programmable Superconducting Processor**, 2019, research.google:48651 [7].report of quantum supremacy on the task of

**sampling the output of a pseudo-random quantum circuit**, that takes about 200 seconds on the**53-qubit**Sycamore superconducting processor while classical state-of-the-art classical algorithms would take 10.000 years.“Up to 43 qubits, we use a Schrödinger algorithm, which simulates the evolution of the full quantum state; the Jülich supercomputer (with 100,000 cores, 250 terabytes) runs the largest cases. Above this size, there is not enough random access memory (RAM) to store the quantum state. For larger qubit numbers, we use a hybrid Schrödinger–Feynman algorithm running on Google data centres to compute the amplitudes of individual bitstrings. […] Although it is more memory-efficient, the Schrödinger–Feynman algorithm becomes

**exponentially more computationally expensive**with increasing circuit depth.”“We have performed random quantum circuit sampling in polynomial time using a physically realizable quantum processor (with sufficiently low error rates), yet no efficient method is known to exist for classical computing machinery.”

“Interaction in quantum systems can spread initially localized quantum information into the many degrees of freedom of the entire system. Understanding this process, known as

**quantum scrambling**, is the key to resolving various conundrums in physics.”“We show that while operator spreading is captured by an efficient classical model, operator entanglement requires exponentially scaled computational resources to simulate. These results open the path to studying complex and practically relevant physical observables with near-term quantum processors.”

“Analogous to classical chaos, scrambling manifests itself as a

**“butterfly effect”**, wherein a local perturbation is rapidly amplified over time. […] Quantum scrambling is enabled by two different mechanisms: operator spreading and operator entanglement.”“entanglement in the space of quantum operators is the key to computational complexity of quantum observable.”

To read again ;)

- Babbush et al.,
**Focus Beyond Quadratic Speedups for Error-Corrected Quantum Advantage**, 2021, research.google:49747.“We conclude that

**quadratic speedups will not enable quantum advantage**on early generations of such fault-tolerant devices unless there is a significant improvement in how we realize quantum error correction. […] find that quartic speedups look significantly more practical.”The central issue is that quantum error correction and the device operation time introduce significant constant factor slowdowns to the algorithm runtime.

The comparison with parallel classical resources is particularly damning for quantum computing, and unfortunately many quadratic quantum speedups (especially those leveraging amplitude amplification) apply to problems that are highly parallelizeable.

- McClean et al.,
**What the foundations of quantum computer science teach us about chemistry**, 2021, research.google:50415 [90].“Leveraging results that quantum computers cannot outpace the physical world, we build to the controversial stance that some chemical problems are best viewed as problems for which no algorithm can deliver their solution in general, known in computer science as undecidable problems. This has implications for the predictive power of thermodynamic models and topics like the ergodic hypothesis. However, we argue that this perspective is not defeatist, but rather helps shed light on the success of existing chemical models like transition state theory, molecular orbital theory, and thermodynamics as models that benefit from data.

To read again ;)

The first demonstration of a

**provable exponential advantage in learning about quantum systems**that is robust even on today’s noisy hardware.Combines quantum computing and quantum sensing to squeeze out more accuracy when measurement quantum systems.

Recipe: Entangle the multiple samples of the measurement (by transducing data from a physical system to a stable quantum memory) and process by a quantum agent: quantum PCA, quantum learning.

experiments with up to 40 superconducting qubits and 1300 quantum gates

See also Tazhigulov (2022) [123] about reaching quantum advantage for modelling (real) physical problems.

## Quantum Chemistry¶

### Experiments¶

- O’Malley et al.,
**Scalable Quantum Simulation of Molecular Energies**, 2016, research.google:44815 [99].first electronic structure calculation performed on a quantum computer

*without exponentially costly precompilation*, on an array of**3 superconducting qubits**to compute the energy surface of molecular hydrogen using two distinct quantum algorithms: (1) unitary coupled cluster method using the variational quantum eigensolver (2) canonical quantum algorithm for chemistry, which consists of Trotterization and quantum phase estimationresults within chemical accuracy of the numerically exact result

- Hempel et al.,
**Quantum Chemistry Calculations on a Trapped-Ion Quantum Simulator**, 2018, research.google:46839 [60].experimental implementation of the variational quantum eigensolver algorithm to calculate the molecular ground-state energies of two simple molecules - molecular hydrogen and lithium hydride - on a trapped-ion quantum hardware using up to

**4 qubits**first multi-ion quantum simulation of quantum chemistry

details two different encodings, trick to circumvent algorithmic unstability during optimization, LiH expensive in terms of runtime

further investigations needed: in mitigation of errors or error suppression, in reducing number of required measurements, for reducing the circuit depth

- Arute et al.,
**Hartree-Fock on a Superconducting Qubit Quantum Computer**, 2020, research.google:49057 [8].quantum modelling of the binding energy of \({\rm H}_6\), \({\rm H}_8\), \({\rm H}_{10}\) and \({\rm H}_{12}\) chains as well as the isomerization of diazene on a superconducting circuit made of up to

**12 qubits**, with a parameterized ansatz circuits realizing the Givens rotation approach to free fermion evolution, variationally optimized to prepare the Hartree-Fock wavefunction, using error-mitigation strategies based on \(N\)-representability

- Tazhigulov et al.,
**Simulating Challenging Correlated Molecules and Materials on the Sycamore Quantum Processor**, 2022, research.google:51198 [123].“With strong quantum advantage demonstrated in artificial tasks,

**we examine how such advantage translates into modeling physical problems**, and in particular, strongly correlated electronic structure.”simulate static and dynamical electronic structure on a superconducting quantum processor for two representative correlated electron problems, on up to

**11 qubits**with up to 780 two-qubit gates: the nitrogenase iron-sulfur molecular clusters, and α-ruthenium trichloride, a proximate spin-liquid materialrun on the best-performing qubits of Google’s 53-qubit Weber processor based on the Sycamore architecture

“Qualitatively correct features in the spin structure, excited-state spectrum, and heat capacity can be obtained. However, to achieve this, implemented circuits need to be obtained with the help of classical recompilation and the data require significant processing. Unfortunately, these steps raise questions with regard to effectively simulating more classically difficult systems.”

The main limitation in the experiments is the two-qubit gate count: simulations with more than 100 gates are not successful.

### Algorithms¶

- Kivlichan et al.,
**Quantum Simulation of Electronic Structure with Linear Depth and Connectivity**, 2018, research.google:46718.proposes an arrangement of qubits to reduce cost of algorithms for practical

*connectivities between qubits*, assuming only a minimal, linearly connected architectureapplies both to variational and phase-estimation-based simulation of quantum chemistry

- Berry et al.,
**Qubitization of Arbitrary Basis Quantum Chemistry Leveraging Sparsity and Low Rank Factorization**, 2019, research.google:47849.proposes a method to reduce the gate complexity by taking advantage of structure in the Coulomb operator

applied to simulation of the FeMoco molecule (relevant to Nitrogen fixation)

- O’Brien et al.,
**Efficient Quantum Computation of Molecular Forces and Other Energy Gradients**, 2021, research.google:50837 [102].introduces new quantum algorithms for computing molecular energy derivatives with significantly lower complexity than prior methods

concludes that calculation of forces on a single nuclei may be of similar cost to estimating energies of chemical systems

- McClean et al.,
**Discontinuous Galerkin Discretization for Quantum Simulation of Chemistry**, 2020, research.google:48291.proposes a method to reduce the costs (in terms of number of integrals) of Gaussian and molecular orbital discretizations in electronic structure calculations

enables to optimize the use of quantum algorithms

- Huggins et al.,
**Efficient and Noise Resilient Measurements for Quantum Chemistry on Near-Term Quantum Computers**, 2021, research.google:48383.previous bounds on the measurement time required by variational algorithms have suggested that the application of these techniques to larger molecules might be infeasible

presents an optimized measurement strategy

provides numerical estimations for calculation of ground-state energies of strongly correlated electronic systems

- Su et al.,
**Fault-Tolerant Quantum Simulations of Chemistry in First Quantization**, 2021, research.google:50356.compile, optimize, and analyze the finite resources required to implement two

**first quantized quantum algorithms**for chemistry, compare to more commonly studied algorithms in second quantizationqubitized algorithm will often be more practical than the interaction-picture algorithm

- Goings et al.,
**Reliably Assessing the Electronic Structure of Cytochrome P450 on Today’s Classical Computers and Tomorrow’s Quantum Computers**, 2022, research.google:51132.Both classical and quantum resource estimates suggest that simulation of CYP models at scales large enough to balance dynamic and multiconfigurational electron correlation has the

**potential to be a quantum advantage problem**and emphasizes the important interplay between classical computations and quantum algorithms development for chemical simulation.

## Quantum Computation¶

### Linear System Solvers¶

- Costa et al.,
**Optimal Scaling Quantum Linear Systems Solver via Discrete Adiabatic Theorem**, 2022, research.google:50899.“Recently, several approaches to solving linear systems on a quantum computer have been formulated in terms of the quantum adiabatic theorem for a continuously varying Hamiltonian. […] Here, we prove a rigorous form of the adiabatic theorem that bounds the error in terms of the spectral gap for intrinsically discrete-time evolutions. In combination with the qubitized quantum walk, our discrete adiabatic theorem gives a speed-up for all adiabatic algorithms.”

Provides an overview table over “The history of the lowest-scaling algorithms for solving linear systems of equations on a quantum computer.”

### Variational Quantum Algorithms¶

- Rieffel et al.,
**XY-mixers: analytical and numerical results for QAOA**, 2020, research.google:49033.“This work explores strategies for enforcing hard constraints by using XY-hamiltonians as the mixer.”

“In this work, we develop intuitive constructions for a large class of these [optimization] algorithms based on connections to simple dynamics of quantum systems, quantum walks, and classical continuous relaxations. We focus on developing a language and tools connected with kinetic energy on a graph for understanding the physical mechanisms of success and failure to guide algorithmic improvement. This physical language, in combination with uniqueness results related to unitarity, allow us to identify some potential pitfalls from kinetic energy fundamentally opposing the goal of optimization. […] Kinetic energy and graph Laplacian perspectives provide new insights to common initialization and optimal solutions in QAOA as well as new methods for more effective layerwise training. […] Connections to classical methods of continuous extensions, homotopy methods, and iterated rounding suggest new directions for research in quantum optimization.”

“VQAs have now been proposed for essentially all applications that researchers have envisioned for quantum computers, and they appear to the best hope for obtaining quantum advantage.”

Provides an overview of

*ansätze*(e.g.*hardware efficient*,*variational hamiltonian*,*unitary coupled cluster*etc.), gradient evaluation, optimizers, applications (chemistry and materials science, optimization, mathematical solvers, machine learning).Lists challenges faced by VQA: trainability (

*barren plateaus*), efficiency, accuracy.“QAOA may be a good candidate VQA to find usage in the fault-tolerant era, albeit with caveats about the overhead.”

### QML¶

- Verdon et al.,
**Learning to learn with quantum neural networks via classical neural networks**, 2019, research.google:.“One such challenge is finding good parameter initialization heuristics that ensure rapid and consistent convergence to local minima of the parameterized quantum circuit landscape. In this work, we train classical neural networks to assist in the quantum learning process, also know as meta-learning, to rapidly find approximate optima in the parameter landscape for several classes of quantum variational algorithms.”

- Broughton et al.,
**TensorFlow Quantum: A Software Framework for Quantum Machine Learning**, 2020, research.google:49371.“We introduce TensorFlow Quantum (TFQ), an open source library for the rapid prototyping of hybrid quantum-classical models for classical or quantum data. […] We illustrate TFQ functionalities via several basic applications including supervised learning for quantum classification, quantum control, simulating noisy quantum circuits, and quantum approximate optimization. Moreover, we demonstrate how one can apply TFQ to tackle advanced quantum learning tasks […].”

- Huang et al.,
**Power of data in quantum machine learning**, 2021, research.google:49725.This paper is about

**learning quantum models**.Data can elevate classical [machine learning] models to rival quantum models, even when the quantum circuits generating the data are hard to compute classically.

Following these constructions, in numerical experiments, we find that a variety of common quantum models in the literature perform similarly or worse than classical ML on both classical and quantum datasets due to a small geometric difference.

With the large geometric difference endowed by the projected quantum model, we are able to construct engineered datasets to demonstrate large prediction advantage over common classical ML models

### Error Correction¶

- Google Quantum AI,
**Suppressing quantum errors by scaling a surface code logical qubit**, 2023, ai.googleblog.com, QEC milestone [2].experimentally demonstrated prototype of quantum error correction i.e. error rate of logical qubit is lower for a code involving more physical qubits than the reference “We want the higher protection offered by QEC to outweigh the increased opportunities for errors as we increase the number of qubits.”

uses a 9-qubit distance-5 (d = 5)

**surface code**““Data” qubits on the vertices make up the logical qubit, while “measure” qubits at the center of each square are used for so-called “stabilizer measurements.” These measurements tell us whether the qubits are all the same, as desired, or different, signaling that an error occurred, without actually revealing the value of the individual data qubits.”

error-correction using

**two different decoders**: belief-matching, an efficient combination of belief propagation and minimum-weight perfect matching ; and tensor network decoding, a slow but accurate approximate maximum-likelihood decoder.“Our target remains achieving logical error rates of 1 in 106 or lower per cycle of QEC.”