This document was suggested in the Whatsapp group of the Quantum Computing Incubator 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. This device has been significantly improved and the article is super interesting. Reason for this blog post. I hope you enjoy it :)

So, this is a joint work between an artificial intelligence and a human. It's a brick to read. Reason for wich it became a glossary, the intention is to invite you to generate a conversation about it, so that you can please comment, correct, or in general tell us what you think. Understand that I am an amateur studying and writing this, but carefully following the statements that were made during the construction of this publication, and for the same reasons, this document will continue to be updated as it goes deeper, the purpose is to reach the point where a person with average or below average knowledge as I consider myself, can better understand or broaden your panorama on the computational and scientific bases that are proposed here.

We will start with the answers given by Google's NotebookLM [1] and then those of OpenAI's GPT chat

---NotebookLM:

Summary

This scientific paper describes improvements in Quantinuum's quantum computer H2, reaching 56 qubits with arbitrary connectivity and a two-qubit gate fidelity of 99.843%. The study focuses on the difficulty of classically simulating the sampling of quantum random circuits (RCS) executed on this enhanced hardware. The empirical results show that the computational power of H2 for RCS is mainly limited by the number of qubits, not by fidelity or speed, exceeding the capabilities of current classical algorithms. Experimental evidence and numerical simulations are presented that compare the fidelity of H2 with classical tensor network-based simulation techniques, confirming a clear quantum advantage at a scale that seems to be beyond the reach of current classical computing. The paper concludes with the promise of the QCCD architecture for the scalability of quantum computers and its growing computational advantage as the number of qubits increases.

Key themes:

Random quantum circuits

Classic Simulation

Quantum computing

Gate fidelity

Advantage

Study Guide: Computational Power of Random Quantum Circuits in Arbitrary Geometries

Summary

This study guide explores the computational power of quantum random circuits (RCS) in various geometries, focusing on the architecture of Quantinuum's quantum computer H2. The study analyzes the complexity of the classic RCS simulation, highlighting the influence of circuit geometry on the difficulty of the simulation. Complexity density is explored as a metric to quantify the difficulty of the simulation, comparing random geometries with 2D geometries.

The paper also presents experimental results of RCS execution in H2, including fidelity estimates using mirror benchmarking techniques and linear cross-entropy. The details of the implementation are discussed, including gate benchmarking and handling memory errors during ion transport. In addition, approximate simulation methods, such as the density matrix renormalization group (DMRG), are explored, evaluating their ability to simulate RCS with achievable fidelities on current hardware.

***

Glossary of Key Terms

Quantum Random Circuit (RCS): A quantum circuit composed of randomly chosen gates from a specific set of gates.

H2 Quantum Computer: A trapped-ion quantum computer developed by Quantinuum.

Complexity Density: A metric that quantifies the fraction of qubits that contribute to the difficulty of simulating a circuit.

Circuit Geometry: A pattern of connectivity between qubits in a quantum circuit.

Mirror Benchmarking: A technique for estimating the fidelity of a quantum circuit by running the circuit followed by its inverse.

Linear Cross Entropy (FXEB): A metric for evaluating fidelity in RCS experiments by comparing the probability distribution of measured bitstrings with the ideal distribution.

Memory Error: Decoherence or errors introduced during the transport of ions in a quantum computer of trapped ions.

Density Matrix Renormalization Group (DMRG): An algorithm for approximating the ground state of a many-body quantum system and simulating the evolution of quantum circuits.

Bond Dimensionality: A parameter that controls the amount of entanglement that an MPS representation of a quantum state can capture.

2Q Entanglement Gate: Quantum gate that generates entanglement between two qubits.

Quantum Advantage: The ability of a quantum computer to perform a task that is untreatable by classical computers.

***

Frequently Asked Questions about Quantinuum's H2 Quantum Computer and Quantum Random Circuit Simulation

1. What is the H2 quantum computer and how is it different from other quantum computers?

The H2 is a quantum computer from Quantinuum based on the "Charge-Coupled Quantum Computer" (QCCD) architecture. Unlike other architectures, the H2's QCCD allows for arbitrary connectivity between its qubits, meaning that any pair of qubits can interact directly without the need for logical exchanges. This is accomplished by physically moving the trapped ions that represent the qubits inside the device.

2. How do you run quantum circuits with arbitrary connectivity on the H2?

H2 uses a combination of "split/merge," "displacement," and "exchange" operations to move ions and place qubits that need to interact in adjacent gate zones. This process is repeated for each layer of the circuit, allowing the execution of circuits with complex interaction patterns that are not possible in architectures with limited connectivity.

3. What are Quantum Random Circuits (RCAs) and why are they important?

RCAs are quantum circuits where gates are chosen randomly. They are important because their classical simulation is thought to be extremely difficult for circuits deep enough and with a considerable number of qubits. This is because RCAs generate highly entangled quantum states that cannot be efficiently represented with classical methods. Thus, the successful execution of RCA on a quantum computer can demonstrate the "quantum advantage" over classical methods.

4. How is the fidelity of RCAs assessed in H2?

The fidelity of RCAs in H2 is estimated by two main methods:

Mirror Benchmarking (MB): A "mirror" circuit is constructed by reversing the gates of the original RCA circuit in the middle. The probability that the final state matches the initial state (MB return probability) is used as an estimate of fidelity.

Linear cross-entropy (FXEB): The cross-entropy between the probability distribution of the results obtained in H2 and the ideal distribution of the noise-free circuit is calculated. An FXEB close to 1 indicates high fidelity.

5. What are the challenges to classically simulate RCA of random geometries?

The classical RCA simulation is based on the representation of the circuit as a tensor network (RT) and the contraction of this network to obtain the probability amplitude of a specific state. The difficulty of RT contraction is related to the "effective number of qubits" (Nd,N), which reflects the computational cost. For RCA with random geometries, Nd,N grows rapidly with the depth of the circuit, saturating the simulation cost of a full-state vector.

6. What methods are used to classically simulate RCA and what are its limitations?

Exact contraction of RT: This method is feasible for shallow circuits, but quickly becomes unfeasible as the depth and number of qubits increases due to the exponential growth of computational cost.

MPS Simulation (DMRG): This method approximates the quantum state with a matrix product state (MPS) and evolves it over time. While it is more efficient than exact contraction, its fidelity decreases as the depth of the circuit and the amount of entanglement increases. In addition, significant computational resources are required to achieve fidelity comparable to that of H2 for deep circuits.

7. How does choosing the two-qubit gate affect the classical simulation difficulty?

In general, two-qubit gates with a higher Schmidt range (such as iSWAP) lead to circuits that are more difficult to simulate than lower-range gates (such as UZZ). This is because higher-range gates generate more entanglement, which increases the complexity of the tensor network.

8. What implications do the H2 results have for the "quantum advantage"?

The H2 results demonstrate the ability of the QCCD architecture to run RCA with high fidelity and arbitrary connectivity. The difficulty of classically simulating these circuits, especially as qubit count and depth increase, suggests that H2 could achieve the "quantum advantage" in RCA execution. Importantly, however, the definitive proof of "quantum advantage" requires a thorough comparison with the best classical algorithms available.

***

Quantinuum H2 improvements that exceed classic limitations

The hardware improvements in the Quantinuum H2 quantum computer that allow it to overcome classical limitations focus on two main aspects:

Arbitrary connectivity: Unlike quantum computing architectures with fixed qubit locations, the H2's QCCD architecture allows for arbitrary connectivity between all 56 qubits. This means that any pair of qubits can be run on a two-qubit gate layer without the need for logical SWAP gates. This feature is critical for running circuits with high connectivity geometries, which are much more difficult to simulate classically. [1-4]

High gate fidelity: The H2 features a two-qubit gate fidelity of 99.843(5)%. This high fidelity significantly reduces the build-up of errors as the depth of the circuit increases. [1, 5, 6]

These two features combined, arbitrary connectivity and high fidelity, allow the H2 to run quantum random circuits (RCS) that are extremely difficult to simulate classically, even at relatively shallow circuit depths. [1, 3, 5]

Impact of connectivity on classical simulation

Classical quantum circuit simulation relies heavily on tensor network (TN) techniques. The complexity of TN contraction, which determines the computational cost of simulation, is highly dependent on the connectivity of the circuit. Circuits with higher connectivity are more difficult to simulate classically because they require the formation of larger intermediate tensors during contraction. [3, 7, 8]

The H2, with its arbitrary connectivity, allows the execution of circuits with random geometries. These circuits exhibit a constant asymptotic complexity density, which means that the cost of classical simulation does not decrease as the number of qubits for a fixed depth increases. This is in contrast to 2D circuits, where the complexity density decreases with the number of qubits, making them easier to simulate. [9, 10]

Impact of High Fidelity on Classic Simulation

The high fidelity of the gates in the H2 also plays a crucial role in overcoming the classic limitations. Approximate TN simulations, such as those based on the Density Matrix Renormalization Group (DMRG) algorithm, rely on the compression of quantum states to make them manageable with the classical resources available. However, this compression introduces errors that increase with the depth of the circuit. [7, 11, 12]

The high fidelity of the H2 gates makes it difficult for approximate simulation methods to compete with the fidelities experimentally achieved for highly entangled states. As the fidelity of quantum gates increases, classical compression-based simulators become less effective. [12, 13]

In summary, the Quantinuum H2's hardware improvements, including arbitrary connectivity and high-fidelity gates, allow it to perform random quantum circuit sampling tasks that are extremely challenging for classical algorithms. These improvements represent a significant advance in the pursuit of a quantum computational advantage.

***

Questionnaire

Answer the following questions in 2-3 sentences each:

1. What is a quantum random circuit (RCS) and what is its importance in the context of quantum computing?

2. It describes the architecture of the quantum computer H2 and how it allows the execution of circuits with arbitrary geometries.

3. What is complexity density and how is it used to quantify the difficulty of simulating quantum circuits?

4. Explains the difference in complexity density between circuits with random geometries and 2D geometries.

5. It describes mirror benchmarking and how it is used to estimate the fidelity of quantum circuits.

6. What is linear cross-entropy (FXEB) and how does it provide a measure of fidelity in RCS experiments?

7. It explains the concept of memory error in the context of trapped-ion quantum computers and how it is mitigated in H2.

8. What is the Density Matrix Renormalization Group (DMRG) and how can it be used to simulate quantum circuits?

9. It describes the challenges of rough RCS simulation using DMRG, particularly in relation to bond dimensionality.

10. How does the choice of the two-qubit entanglement gate (2Q) influence the complexity of quantum circuit simulation?

Keys to Answers

1. An RCS is a quantum circuit with randomly chosen gates from a specific set of gates. Its importance lies in its ability to generate complex quantum states that are difficult to simulate classically, providing proof of quantum advantage.

2. H2 is based on a racetrack-shaped trap of trapped ions, where each qubit is encoded in an ion of Yb+. The architecture allows for arbitrary ion transport and rearrangement, allowing for the execution of circuits with any connectivity pattern.

3. Complexity density is a measure of the fraction of qubits that contribute to the difficulty of simulating a circuit. It quantifies the scalability of the classical resources needed to simulate a quantum circuit.

4. Circuits with random geometries exhibit a higher complexity density compared to circuits with 2D geometries. This implies that circuits with random geometries are significantly more difficult to simulate classically for a given size and depth.

5. Mirror benchmarking involves running a circuit followed by its inverse, with the expectation of recovering the initial state. Fidelity is estimated from the probability of returning the initial state, providing a measure of the accuracy of the circuit.

6. FXEB is a metric that compares the probability distribution of the measured bitchains of a noisy quantum circuit to the ideal distribution. A higher FXEB indicates higher fidelity, suggesting that the circuit behaves closer than expected.

7. Memory error refers to decoherence or errors introduced during ion transport in a trap. In H2, these errors are mitigated by using sympathetic cooling and optimizing transport sequences to minimize operation times.

8. DMRG (Density Matrix Renormalization Group) is a variational algorithm for approximating the ground state of a quantum system of many bodies. In the context of quantum circuit simulation, DMRG can approximate the quantum state during circuit evolution, allowing the simulation of larger circuits than is possible with exact methods.

9. The main challenge in rough RCS simulation using DMRG lies in choosing a suitable bond dimensionality for the MPS representation of the quantum state. A higher link dimensionality allows more entanglements to be captured, but significantly increases computational complexity.

10. The choice of the 2Q gate impacts the complexity of the simulation. Doors with a higher Schmidt range, such as the iSWAP gate, generally lead to circuits that are more difficult to simulate than doors with a lower Schmidt range, such as the UZZ(π/2) gate.

***

How do you generate a Cd,N random geometry circuit?

A random geometry circuit, denoted as Cd,N, is constructed using a regular random graph as a basis. This process is described in detail in section A4 of document "2406.02501v3" and is summarized in the following steps:

1. Graph selection: You start by selecting a random graph d-regular, denoted Gd,N, with N vertices. A d-regular graph is one in which each vertex has exactly d edges connected to it.

2. Vertex-Qubit Correspondence: Each vertex of the graph Gd,N is associated with a qubit in the circuit Cd,N.

3. Assignment of Two-Qubit Gates (2Q): Each edge of the graph is associated with a two-qubit gate in the circuit. The native two-qubit gate in Quantinuum's H2 quantum computer is the UZZ(π/2) = e−i(π/4)Z⊗Z gate, which is used for all (N×d/2) 2Q gates in the circuit.

4. Edge coloring: To determine the order of the 2Q gates in the circuit, a coloration of edges of the graph Gd,N is performed using d colors. An edge coloration is "proper" if there are no two edges of the same color that share a common vertex. In other words, each color defines a set of 2Q gates that can be run in parallel.

5. Circuit Layering: Each color in the edge coloration is associated with a layer of the Cd,N circuit. The edges of a given color are translated into 2Q gates on the corresponding layer of the circuit. Therefore, each layer of the circuit contains N/2 2Q gates, which can be applied simultaneously thanks to the flexible connectivity of the QCCD architecture.

6. Insertion of Single-Qubit (1Q) Gates: To complete the construction, layers of 1Q gates are inserted between each layer of 2Q gates, as well as before initialization and after measurement of the qubits. Each 1Q gate is randomly chosen from the SU(2) group.

In summary, the generation of a random geometry Cd,N circuit involves the transformation of a d-regular random graph into a quantum circuit, where the vertices represent qubits, the edges represent 2Q gates, and the coloration of edges determines the stratification of the circuit.

Figure 2 of the paper provides an illustration of this process for a 3-depth circuit with 6 qubits. Figure A14, in the appendices section, also gives a visual representation of the construction of a Cd,N circuit from a graph.

---CHAT OPENAI:

The prompt for Chat GPT was mainly to:

1. Assemble the item

2. Ask for definitions from the list of key concepts underlined during the reading.

3. Ask for a brief summary of the article.

4. Make a short writing with some conclusions taken from the text. Then I asked the chat to solve doubts, edit or remove what was not in the document or was not verified in the online data.

Neither item (3) nor (4) is presented because NotebookLM gave us a very complete summary. However, other questions that are considered important are attached here:

Do you want complexity density to tend to zero or one and why? It depends on the context. In general, if the complexity density tends to one, it means that the problem is highly complex and difficult to simulate classically. If it tends to zero, the problem is more easily solved by classical techniques.

What does it mean that at sufficient depth the optimal TN contraction proceeds in a time-like direction, but that at low depths it has a non-time-like behavior? At high depths, tensor contraction follows a clear time-like direction, meaning it aligns with the natural evolution of the system. At shallow depths, the circuit structure allows for different contraction strategies that do not necessarily follow a temporal order. 

How are F, F_EXB, F_MB and F_GC related? These metrics represent different fidelities in the quantum system. F is probably the overall fidelity, while F_EXB, F_MB and F_GC reflect specific loyalties under different evaluation methods.

What is the advantage of Yb? Ytterbium (Yb) is used in quantum computing for its stable electronic transitions, low decoherence, and laser cooling capability, which improves the accuracy of quantum gates.

What is the advantage of crystal properties? Ionic crystals, such as those in Yb-Ba, allow qubits to be stored and manipulated with greater stability, minimizing errors due to unwanted interactions and improving connectivity between qubits.

What is the importance of low-density parity check codes? They are error-correcting codes that help improve the fidelity of quantum computing. They can reduce the resources required to implement error correction in quantum hardware.

What is a two-ion crystal? It is a configuration where two ions are trapped and coupled in an ionic crystal, allowing the implementation of quantum operations with greater stability and lower noise.

What do you mean by "sampling uniformly at random"? It means taking samples randomly but with an even distribution, ensuring that every possible result has an equal chance of being selected.

What is a "non-identity Pauli error"? It refers to an error in the quantum system that is not the identity (I) in the Pauli base. For example, errors of type XX, YY, or ZZ, which represent bit-flip, phase-flip, or both errors.

What does "good edge expansion properties" mean? It refers to the fact that a graph structure has good connectivity between nodes, which improves entanglement propagation and efficiency in quantum circuits. 

What do you mean by "sequence length"? It probably refers to the number of gates or steps in a quantum circuit, which influences the fidelity and difficulty of simulation.

What are the types of memory errors that are considered? Decoherence: Loss of quantum information due to interactions with the environment. Storage errors: Variations in the state of a qubit due to thermal noise or field fluctuations. Transport errors: Failures when moving ions or performing operations on specific architectures.

Explanation of the impact of gate choice on simulation difficulty At high depth, tensor contraction follows a clear time direction, with computational costs similar to state vector simulation. At shallow depth, optimal contraction does not follow this time structure, and the simulation cost can be reduced by using low-complexity quantum gates (such as UZZ(π/2) instead of iSWAP). The choice of gates affects the simulation difficulty, as some allow for more efficient decompositions and reduce computational cost.

And here is the glossary (in development):

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