This document was suggested in the Whatsapp group of the Quantum Computing starter group of the National University of Colombia. In this paper, called The Computational Power of Random Quantum Circuits in Arbitrary Geometries (refered to as the document: 2406.02501v3), Quantinuum and other major tech companies share the results of their latest version of the H2 quantum computer.

For now, I will share a key terms glossary. But I will post a better description about the paper ;)

Glossary of key terms

A Nonlinear Least-Squares Fit of L(N): A nonlinear least squares fit applied to the function L(N), which describes the difficulty of simulating a quantum circuit as a function of the number of qubits N.

Aggregate Bootstrapping and Double Bootstrapping (r = 4000 resamples): Statistical methods for estimating the distribution of a dataset by repeated sampling. Double bootstrapping uses two levels of resampling to improve estimation.

Ansatz Constrained: An ansatz is an initial assumption or approximation about the shape of a solution. Ansatz constrained means that this approximation is limited by resources such as memory or computational time. Example: In tensor network simulations, you can limit the size of the bond dimension of the ansatz to handle larger systems.

Asymptotic Complexity Density: Measures how difficult it is to simulate a quantum circuit based on the number of qubits and the depth of the circuit.

Asymptotic Expansion Properties: In large-scale simulations, when N→∞ (large number of qubits), asymptotic expansion properties help to model the behavior of the system at the boundary, allowing to analyze scalability patterns and the difficulty of simulating larger quantum circuits.

Asymptotic Hardness: Refers to the difficulty of simulating a quantum system as the number of qubits and the depth of the circuit increase indefinitely.

B

Binned: The process of grouping continuous values into intervals or "bins" to analyze probability distributions.

Bipartite Entanglement: Measures how entangled two parts of a quantum system are. Example: A state ∣Ψ⟩=12(∣00⟩+∣11⟩) has maximum bipartite entanglement, since the qubits are fully correlated.

Bisecting Surface: An imaginary surface that divides a tensor network into two approximately equal parts. It is used to analyze and calculate entanglement entropy and optimize the contraction of tensor networks.

Bitstrings: Sequences of bits generated as output of a quantum circuit.

Bond Dimension: In tensor networks, it is the maximum size of the internal indices between tensors, determining how much information can flow through connections in the network.

Brute-force TN Methods: Classical quantum circuit simulation methods that use brute force in tensor networks to calculate output probabilities.

C

Cd,N: Represents a quantum circuit with N qubits and dd layers of quantum gates. It is also used to denote the complexity density of the circuit.

Classical Simulability: The ease with which a quantum system can be simulated by a classical computer.

Closed-Simulation Approach: Simulation of a complete quantum circuit without approximations.

Coherent Errors: Errors in a quantum system that occur systematically and cumulatively due to imperfections in the quantum gates or in the calibration of the system.

Computational Cost: Computational resources required to simulate a quantum circuit.

Complexity Density: The proportion of qubits in a circuit that actually contribute to computational complexity.

Constituent Operation: An individual operation that is part of a larger sequence in the execution of a quantum circuit.

Contraction Cost for TN: Computational cost of contracting a tensor network.

Contraction Width (W): The size of the largest tensor generated during the contraction of a tensor network.

Cost Sublinear in S: When the computational cost grows more slowly than the number of SS samples.

Cross-Entropy Benchmarking (FXEB): A metric that compares theoretical probabilities and probabilities measured in quantum circuits to assess fidelity.

D

D (Depthness of the Circuit): Depth of the quantum circuit, i.e. the number of layers of quantum gates applied.

Distribution of Binned Output Probabilities: Representation of the output probabilities of a quantum circuit organized in discrete intervals (bins), facilitating statistical analysis.

And

Effective Error per 2Q Gate (ε): Error accumulated by each two-qubit gate in quantum hardware.

Estimated Simulation Fidelity: A measure of how close a classical simulation is to the ideal execution of a quantum circuit.

Exponential Tail: Part of a probability distribution that decreases exponentially, indicating the low occurrence of extreme events.

Extensive Caching: Intensive use of caching to reduce computation time in quantum simulations.

F

Faster Clock Speed: The speed of the processor clock on quantum hardware.

FLOP's Operations (assuming no memory constraints): Number of floating-point operations required for a quantum simulation without memory constraints.

G

Gate Fidelity: A measure of how accurately a quantum gate operates in a system.

Gd,N (Random d-Regular Graph on N Nodes): A random d-regular graph on N nodes that defines connectivity in a quantum circuit.

Global Microwave Rotations: Quantum rotations applied to all qubits by microwave pulses.

H

Haar Gates: Random quantum rotations applied to qubits.

Haar-Random SU(2): Random quantum operation within the SU(2) group.

Higher Connectivity: Greater interconnection between qubits, which influences simulation complexity.

Highly-Entangled States: Quantum states with a high level of entanglement.

Hyperfine States: Energetic states of an ion that result from the interaction between nuclear and electronic magnetic moments.

I

Isoperimetric Number: A parameter that describes the relationship between the size of a set of nodes and the minimum number of connections that must be cut to separate it from the rest of the system. In tensor networks, this influences the difficulty of simulation.

L

Light Cone Based Contraction Ordering: Strategy to optimize the contraction of tensor networks based on the causality of the light cone, reducing computational complexity.

Logistic (Sigmoid) Function: An S-shaped mathematical function used to model smooth transitions between two states, common in data analysis and machine learning.

Low-Entanglement Partitions: Partitions of a quantum system where the parts are weakly correlated.

M

Machine Control System (FPGA): FPGA-based system for controlling quantum hardware.

Matrix-Product-State (MPS) Ansatz: A method of tensor network compression used to simulate quantum systems with limited entanglement.

Memory-Constrained TN: Tensor network limited by memory constraints.

Minimal Bisecting Surface: Partition surface with the minimum number of cut connections.

N

Native Gate Set: A set of quantum gates implemented directly into the hardware.

Native Perfect Entangler: Native quantum gate that maximizes entanglement between qubits.

Normalized Effective Qubit Number: Effective number of normalized qubits, defined as Cd,N (ecn. 6).

Non-Local Geometry: Connectivity between qubits without spatial restrictions.

Non-Parametric Bootstrap Resampling: A statistical resampling method that does not assume a previous distribution of data, useful for estimating errors and distributions in quantum systems.

Or

One Qubit Gate: A quantum gate that acts on a single qubit.

Overall Fidelity (FMPS): Total fidelity in an MPS-based quantum simulation.

P

Parameterized Entangler UZZ(θ): Quantum gate that introduces entanglement with an adjustable parameter.

Past Causal Cone: A set of events in a quantum system that can affect a specific point in time, used to determine the evolution of the quantum state.

Pauli Errors: Quantum errors modeled by the gates of Pauli X, Y, and Z.

Perfect Entangler UZZ(π/2): Quantum gate that maximizes entanglement.

Porter-Thomas Distribution: Probabilistic distribution of the squared amplitudes of random quantum states.

Q

QCCD (Quantum Charge-Coupled Device) Architecture: Trapped ion architecture for quantum computing.

QEC (Quantum Error Correction): Correction of quantum errors.

Qubit Reuse Algorithm: An algorithm that optimizes the use of qubits in a quantum circuit by reusing them at different stages of computing.

Qubit's Wire: A visual representation of a qubit in a quantum circuit diagram, indicating its evolution over time.

R

Random Geometry: Random connectivity between qubits in a quantum circuit.

RCS (Random Circuit Sampling): A method for generating highly entangled quantum states.

S

Schmidt-Decompose Rank-2-3-4 Tensors: Process of decomposition of tensors into factors of lower range by Schmidt decomposition, useful for reducing complexity in quantum simulations.

Semi-Parametric Bootstrap Resampling: A hybrid resampling technique that combines parametric assumptions with observed data to improve statistical estimation.

Standard Clifford Randomizations: Clifford randomization refers to a technique in quantum computing for minimizing coherent errors by applying Clifford operations randomly before and after a main quantum operation. Clifford operations form a special subset of the unitary group, which includes gates such as H (Hadamard), S, and CNOT, and which have the property of mapping computational states and their entanglements into controlled linear combinations. In the context of randomization, these operations are applied randomly to “mix” the effects of the error, making it easier to model as a stochastic error (incoherent noise). Purpose is to reduce the impact of coherent errors, such as small deviations in quantum gate parameters, and to facilitate the calibration and correction of quantum hardware.

Suppress Boundary Effects: Methods to reduce the impact of boundary conditions in quantum simulations, avoiding bias in the results.

T

TC is Composed of NdNd Original Tensors (Small Edge Boundary): In tensor networks, TCTC represents a structure composed of original NdNd tensors with a small boundary, which affects the shrinkage efficiency.

Time-Like Direction: In a quantum simulation, it refers to the direction in which information flows causally, similar to how time moves in a physical system.

Or

Under-Cover the Data: A situation in which a statistical model does not fully capture variations in the data, which can lead to biased or incomplete estimates.

W

Worst-Case Hardness of Simulating Nd Qubits: Maximum difficulty of simulating a quantum circuit with Nd qubits.

REFERENCES

DeCross M, Haghshenas R, Liu M, Rinaldi E, Gray J, Alexeev Y, Baldwin C, Bartolotta J, Bohn M, Chertkov E, Cline J, Colina J, DelVento D, Dreiling J, Foltz C, Gaebler J, Gatterman T, Gilbreth C, Giles J, Gresh D, Hall A, Hankin A, Hansen A, Hewitt N, Hoffman I, Holliman C, Hutson R, Jacobs T, Johansen J, Lee P, Lehman E, Lucchetti D, Lykov D, Madjarov I, Mathewson B, Mayer K, Mills M, Niroula P, Pino J, Roman C, Schecter M, Siegfried P, Tiemann B, Volin C, Walker J, Shaydulin R, Pistoia M, Moses S, Hayes D, Neyenhuis B, Stutz R and Foss-Feig M. The computational power of random quantum circuits in arbitrary geometries (2024), arXiv:2406.02501v3. https://arxiv.org/abs/2406.02501

CYBERGRAPHY

[1]. NOTEBOOkLM: https://notebooklm.google.com/notebook/2954d414-105c-40b3-8933-ba1854917f5a?_gl=1*k4fmwy*_ga*MTMzMjUxMDYxLjE3MzMzNjgxNjU.*_ga_W0LDH41ZCB*MTczMzM2ODE2NS4xLjAuMTczMzM2ODE2NS42MC4wLjA.&original_referer=https:%2F%2Fnotebooklm.google%23&pli=1

[2]. Chat GPT - OPENAI: https://chatgpt.com/

You can download here the pdf with the figures and descriptions from the paper