004 - Understanding Noise in Near-Term Quantum Computers with Haimeng Zhang: Qiskit Summer School 2024

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https://www.youtube.com/watch?v=w--HZ4QXugA&list=PLOFEBzvs-Vvr-GzDWlZpAcDpki5jUqYJu&type=snipo

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Completed

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Qiskit

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Video Duration

00:47:52

The image shows a diagram and explanation of a quantum circuit, which is described as consisting of unitary gates applied to qubits. The circuit has three main stages:

Qubit initialization: The diagram shows 4 qubits (q0 to q3) initialized in different states (S, H, Z for the first three).

Apply quantum gates: The middle section shows various quantum gates being applied, including basis gates like ECR, ID, RZ, SX, and X, which are noted to form a universal gate set. The diagram illustrates several gates like Rx, T†, Y, and Ry being applied to different qubits.

Measurement: The final stage shows measurement operations applied to all four qubits.

The image also includes a photo of a person identified as Haimeng Zhang, a Quantum Engineer, who appears to be presenting this information.

the mentioned basis gates are:

ECR (Echoed Cross-Resonance Gate): This is a two-qubit gate used for entangling qubits and performing operations like controlled-not (CNOT). It's a key gate in superconducting quantum circuits.

ID (Identity Gate): This is a gate that performs no operation on the qubit. It's useful for aligning operations in a circuit but leaves the qubit unchanged.

RZ Gate: This gate performs a rotation around the Z-axis of the Bloch sphere by a specified angle. It’s a single-qubit gate.

SX Gate: Also known as the square-root of X gate, it’s a single-qubit gate that applies half the rotation of the Pauli-X gate. It can be used to prepare superposition states.

X Gate: The Pauli-X gate flips the state of a qubit (|0⟩ to |1⟩ and |1⟩ to |0⟩), essentially like a classical NOT gate.

The image shows a slide from a lecture about noise in quantum hardware, given by Haimeng Zhang, a Quantum Engineer. The slide discusses several key points:

Quantum circuits are compiled and run physically on quantum hardware.

The performance and power of today's quantum processors are largely limited by noise.

Understanding the limitations of quantum hardware due to noise is important to make the best use of it.

The lecture aims to cover:

Different sources of noise

Where to read noise-related metrics from the IBM Quantum Platform

How these noise factors are experimentally characterized

The slide includes a diagram of what appears to be a quantum processor layout from IBM's Sherbrooke system, showing a grid of interconnected qubits.

To explain how to use the measurement operators for measuring a qubit, let's break it down step-by-step using the formulas provided in the image:

The measurement operators for a single qubit are given as: P₀ = |0⟩⟨0| (measures the probability of the qubit being in state |0⟩) P₁ = |1⟩⟨1| (measures the probability of the qubit being in state |1⟩)

Let's say we have a qubit in state |ψ⟩. To measure this qubit:

For measuring in state |0⟩: Probability = ⟨ψ|P₀|ψ⟩ = ⟨ψ|(|0⟩⟨0|)|ψ⟩

For measuring in state |1⟩: Probability = ⟨ψ|P₁|ψ⟩ = ⟨ψ|(|1⟩⟨1|)|ψ⟩

These probabilities give you the likelihood of measuring the qubit in the respective states.

The sum of these probabilities should equal 1: ⟨ψ|P₀|ψ⟩ + ⟨ψ|P₁|ψ⟩ = 1

After measurement, the qubit collapses to the measured state. If you measured |0⟩, the post-measurement state is: |ψ'⟩ = (P₀|ψ⟩) / √(⟨ψ|P₀|ψ⟩)

Similarly, if you measured |1⟩, the post-measurement state is: |ψ'⟩ = (P₁|ψ⟩) / √(⟨ψ|P₁|ψ⟩)

These formulas allow you to calculate the probabilities of measuring each computational basis state and determine the post-measurement state of the qubit. The actual measurement process in a quantum computer will return either 0 or 1 based on these probabilities.

This slide discusses the sources of noise in a quantum circuit, which are a critical factor affecting the accuracy and reliability of quantum computations. Here's an overview of the key points:

Incoherent errors:

These errors arise from the loss of quantum information, particularly in the form of superposition and entanglement. Incoherent errors often result from environmental interactions that cause qubits to lose their quantum states, a phenomenon called decoherence.

Coherent errors in gates:

Coherent errors result from incorrect Hamiltonian evolution, which means that the quantum gates are not performing their operations exactly as intended. These errors tend to be systematic and can accumulate over time in a quantum circuit.

State Preparation and Measurement (SPAM) errors:

These errors occur during the preparation of qubit states and the measurement process. Inaccuracies in preparing the correct initial quantum state or measuring the final state introduce errors that affect the overall fidelity of the quantum computation.

The image shows a representation of a quantum bit (qubit) using the Bloch sphere model. It explains the density matrix representation for both pure and mixed quantum states.

For a pure state, the density matrix is given as ρ = |ψ⟩⟨ψ|.

For a mixed state, the density matrix is represented as ρ = Σᵢ qᵢ|ψᵢ⟩⟨ψᵢ|, which describes an ensemble of pure states {qᵢ, |ψᵢ⟩}.

The Bloch sphere shown in the image is a geometrical representation of the state space of a qubit. It's a unit sphere where:

The north pole represents the |0⟩ state

The south pole represents the |1⟩ state

Any point on the surface represents a pure state

Any point inside the sphere represents a mixed state

Regarding the states that can be represented on this Bloch sphere:

Pure states: These are represented by points on the surface of the sphere. Any pure state of a qubit can be written as a superposition of |0⟩ and |1⟩, and corresponds to a unique point on the Bloch sphere's surface.

Mixed states: These are represented by points inside the sphere. Mixed states are statistical mixtures of pure states and have less than maximum purity.

The center of the sphere represents the maximally mixed state.

The poles and equator have special significance:

North pole: |0⟩ state

South pole: |1⟩ state

Equator: Equal superpositions of |0⟩ and |1⟩ with varying phases

Any rotation of the Bloch vector (the arrow in the image) corresponds to a unitary operation on the qubit.

I'm going to analyze the image showing the effect of noise on quantum states and operations.

The image depicts a diagram contrasting ideal (noiseless) and noisy quantum systems. It illustrates how noise affects quantum states and operations.

Key elements:

Noise channel representation

Comparison of ideal vs noisy quantum states

Comparison of ideal vs noisy quantum operations

I'll break down each component:

Noise channel:

Represented by a box with a lightning bolt symbol

Input state ρ passes through the noise channel ε

Output state ρ' = ε(ρ)

Quantum states: Ideal: Pure state with ||v→|| = 1 Noisy: Mixed state with ||v→|| < 1

Quantum operations: Ideal: Unitary operations |ψ⟩ → U|ψ⟩ Noisy: Noise channel ε, ρ → ε(ρ)

This diagram effectively illustrates how noise introduces imperfections in quantum systems, transitioning from pure to mixed states and from unitary operations to noisy channels.

I will analyze the information presented in the image about SPAM errors in quantum computing, focusing on the key concepts and data shown.

The image discusses SPAM (State Preparation And Measurement) errors in quantum computing. These errors arise from imperfect state preparation and measurement readout in quantum circuits. The slide explains that in practice, it's difficult to distinguish between state preparation errors and measurement errors.

Key points:

A mix state (e.g., imperfect state preparation) is mathematically described as: P_noisy = A P_ideal Where P_i is the probability of measuring bitstring i, and A_ij is the probability of observing state j while being prepared in state i.

Characterizing these errors at scale is challenging, involving (2^n)^2 elements for n qubits.

The errors are reported on a qubit-by-qubit basis on the backend but could be correlated between qubits.

Two graphs are presented:

A bar chart comparing ideal and noisy distributions of quantum states

A heatmap showing the A matrix for five qubits, illustrating the probabilities of observing different states

The research is based on a paper titled "f" published in PRX Quantum in 2021.

I'm going to analyze the details provided about the IBM Sherbrooke quantum computer system.

The image shows specifications for the IBM Sherbrooke quantum computer, which is part of the OpenQASM 3 platform. Here are the key details:

Number of qubits: 127

EPLG (Error Per Logic Gate): 1.7%

CLOPS (Circuit Layer Operations Per Second): 5K

Status: Online

System region: us-east

Total pending jobs: 25 jobs

Processor type: Eagle r3

Version: 1.4.49

Basis gates: ECR, ID, RZ, SX, X

User's instance usage: 0 jobs

Median ECR error: 6.983e-3

Median SX error: 2.093e-4

Median readout error: 1.370e-2 (highlighted in the image)

Median T1 time: 264.82 μs

Median T2 time: 185.58 μs

The median readout error is highlighted, which relates to the SPAM errors discussed in the previous image. This error rate (1.37%) represents the typical measurement inaccuracy when reading the state of a qubit.

I am going to analyze and explain the image showing a diagram and mathematical representation of noise channels in quantum information theory.

The image depicts a schematic representation of a noise channel in quantum information theory, along with its mathematical formulation. Here's a breakdown of the key elements:

Diagram:

Input state ρ (rho) enters a "Noise channel" represented by a box with the symbol ℰ (script E).

Output state ρ' = ℰ(ρ) emerges from the channel.

A lightning bolt symbol in the box indicates the noisy nature of the channel.

Mathematical representation:

The Kraus representation of the quantum channel is given: ℰ(ρ) = Σ_i K_i ρ K_i^†

The completeness relation is also provided: Σ_i K_i^† K_i = I

Properties of the channel:

Linear

Complete positive

Trace preserving

Additional information:

The image mentions "Further reading: Quantum Channels, John Watrous, IBM Quantum Learning"

https://learning.quantum.ibm.com/course/general-formulation-of-quantum-information/quantum-channels

This representation describes how a quantum state ρ is transformed by a noisy quantum channel. The Kraus operators K_i characterize the channel's effect, and the completeness relation ensures that the channel preserves probability.

explain the properties of linear, completely positive, and trace-preserving quantum channels

To explain these properties, I'll analyze each one in the context of quantum information theory and channel operations:

Linear: A quantum channel ℰ is linear if for any two input states ρ₁ and ρ₂, and any scalar λ: ℰ(λρ₁ + (1-λ)ρ₂) = λℰ(ρ₁) + (1-λ)ℰ(ρ₂) This property ensures that the channel respects superposition of quantum states.

Completely Positive: A channel ℰ is completely positive if it maps positive semidefinite operators to positive semidefinite operators, even when acting on a part of a larger system. Mathematically: For any positive semidefinite operator ρ on a Hilbert space H, (I ⊗ ℰ)(ρ) is also positive semidefinite, where I is the identity operation on an arbitrary ancillary system. This property ensures that the channel preserves the physicality of quantum states, even when entangled with other systems.

Trace Preserving: A channel ℰ is trace preserving if it maintains the trace of the input state: Tr(ℰ(ρ)) = Tr(ρ) for all input states ρ This property ensures that the total probability is conserved, which is crucial for maintaining the normalization of quantum states.

These properties together ensure that quantum channels map valid quantum states to valid quantum states, preserving the fundamental principles of quantum mechanics and probability theory. To explain positive semidefinite operators:

Definition: A Hermitian operator A is positive semidefinite if for any non-zero vector |ψ⟩: ⟨ψ|A|ψ⟩ ≥ 0

Properties:

All eigenvalues are non-negative

Can be expressed as A = B^†B for some operator B

For density matrices ρ, Tr(ρ) = 1 and ρ is positive semidefinite

Importance in quantum mechanics:

Density matrices must be positive semidefinite to represent valid quantum states

Observables (measurement operators) are typically positive semidefinite

Ensures probabilities are non-negative

Role in completely positive channels:

Channels must map positive semidefinite operators to positive semidefinite operators

Preserves the physicality of quantum states throughout information processing

Crucial for maintaining the probabilistic interpretation of quantum mechanics

Mathematical representation: For a matrix A, it is positive semidefinite if and only if: x^† A x ≥ 0 for all complex column vectors x

This property ensures that quantum channels maintain the physical validity of quantum states during information processing and transformations. Importance of positive semidefinite operators in quantum mechanics:

Density matrices:

Represent quantum states

Must be positive semidefinite to ensure physical validity

Observables:

Represent measurable quantities

Eigenvalues correspond to possible measurement outcomes

Must be positive semidefinite to ensure non-negative probabilities

Quantum operations:

Must preserve positive semidefiniteness to maintain physical validity of states

Proof of importance:

Let ρ be a density matrix representing a quantum state.

Positivity: ⟨ψ|ρ|ψ⟩ ≥ 0 for all |ψ⟩

Trace condition: Tr(ρ) = 1

Spectral decomposition: ρ = Σᵢ λᵢ |ψᵢ⟩⟨ψᵢ|, where λᵢ are eigenvalues and |ψᵢ⟩ are eigenvectors

Probability interpretation: P(ψᵢ) = ⟨ψᵢ|ρ|ψᵢ⟩ = λᵢ

For physical validity: 0 ≤ P(ψᵢ) ≤ 1 for all i

Therefore: 0 ≤ λᵢ ≤ 1 for all i

This proves that ρ must be positive semidefinite (all eigenvalues non-negative) to represent a valid quantum state with proper probability interpretation.

The preservation of positive semidefiniteness through quantum operations ensures that the resulting states remain physically valid, maintaining the probabilistic nature of quantum mechanics.

What I'm going to do: I will analyze the image provided and explain the key concepts related to incoherent errors in superconducting qubits, focusing on energy relaxation and the amplitude damping channel.

The image discusses incoherent errors in superconducting qubits, specifically focusing on energy relaxation and the amplitude damping channel. Here's a breakdown of the key points:

Energy relaxation:

Superconducting qubits operate at very low temperatures, typically around 20 mK, with qubit frequencies between 4-6 GHz.

There is stronger decay from the excited state |1⟩ to the ground state |0⟩ compared to excitation from |0⟩ to |1⟩.

Energy relaxation is characterized by the timescale T1.

The probability of a relaxation error is given by 1 - e^(-t/T1), where t is the circuit evolution time.

Amplitude damping channel:

Represented by two Kraus operators: K0 = [[1, 0], [0, √(1-p)]] K1 = √p |0⟩⟨1|

Equivalently: |0⟩ → |0⟩ with probability 1 |1⟩ → |0⟩ with probability p

Bloch sphere representation:

The image shows a Bloch sphere with the ground state |0⟩ at the top and the excited state |1⟩ at the bottom.

The purple cap at the top represents the tendency of the qubit to relax to the ground state due to energy relaxation.

Implications:

The ground state |0⟩ becomes the steady state of the superconducting qubit.

The ground state is largely unaffected by relaxation errors.

This information highlights the challenge of maintaining coherence in superconducting qubits due to energy relaxation, which is an important consideration in quantum computing and error correction strategies.

What I'm going to do: I will analyze the image provided and explain the key concepts related to incoherent errors in quantum systems, specifically focusing on dephasing and the phase damping channel.

The image discusses incoherent errors in quantum systems, with a focus on dephasing and the phase damping channel. Here's a breakdown of the key points:

Dephasing:

Superposition states become classical mixtures due to dephasing.

Dephasing is characterized by the timescale T2.

The probability of a dephasing error is given by 1 - e^(-t/T2), where t is the circuit evolution time.

Effects on quantum information:

Both relaxation (discussed in a previous image) and dephasing errors turn quantum information classical.

Phase flip channel:

Represented by two Kraus operators: K0 = √(1-p) I, where I is the identity matrix K1 = √p Z, where Z is the Pauli-Z matrix

Bloch sphere representation:

The image shows a Bloch sphere with |0⟩ at the top and |1⟩ at the bottom.

The purple vertical slice through the center of the sphere represents the action of the phase damping channel.

This visualization shows how dephasing affects the quantum state by collapsing superpositions along the z-axis.

Implications:

Dephasing preserves the populations (probabilities) of |0⟩ and |1⟩ states.

However, it destroys the phase relationship between these states, which is crucial for maintaining quantum superpositions.

This process effectively reduces the quantum nature of the system, making it behave more classically over time.

Understanding and mitigating dephasing errors is crucial for maintaining quantum coherence in quantum computing systems and for implementing effective quantum error correction strategies. The phase damping channel model provides a mathematical framework for describing and analyzing these errors in quantum information processing.

[1] E. Berg et al., Probabilistic error cancellation with sparse Pauli-Lindblad models on noisy quantum processors. Nature Physics, pages 1-6, 2023.

Pauli noise channels are unital, meaning they map the maximally mixed state to itself.

This image discusses two-qubit errors, specifically crosstalk, in quantum computing. Here's a summary of the key points:

Echoed cross-resonance (ECR) gate: This is equivalent to a CNOT gate up to single-qubit rotations.

Hamiltonian: The image shows the effective Hamiltonian (H_eff) for this system, which includes terms for IX, IZ, ZI, ZX, and ZZ interactions.

ZZ coupling: The ZZ static capacitive coupling results in unwanted crosstalk errors between connected qubits.

Gate duration: Two-qubit gates like ECR typically have longer durations compared to single-qubit gates, by an order of magnitude.

Circuit diagram: The image shows an ECR gate and its equivalent representation using rotations (R_ZX) and an X gate.

the key points:

Coherent errors often result in oscillations in signals.

Coherent errors can build up much more rapidly than incoherent errors.

Some solutions for coherent errors include dynamical decoupling and Twirling (random compiling).

The graph shows "Single-qubit over rotation for d cycles", demonstrating how the measured <Z> (expectation value of the Z operator) oscillates as the circuit depth d increases from 0 to 100.

The oscillatory pattern in the graph visually represents the effect of coherent noise on a single qubit over multiple cycles of operation.

The example is credited to a lecture by Dr. Zlatko Minev at the Qiskit Global Summer School 2023.

The image illustrates different approaches to handling errors in a chain of physically connected qubits, specifically addressing crosstalk and decoherence issues in quantum computing. Here's a breakdown of the key points:

Chain of qubits: The diagram shows a chain of 6 qubits (q0 to q5) connected in a linear arrangement.

Crosstalk: This diagram shows how crosstalk can affect nearby qubits when operations are performed simultaneously. ECR (Echoed Cross Resonance) gates are shown connecting multiple qubit pairs at the same time.

Decoherence: The third diagram demonstrates a technique to mitigate decoherence effects. It uses an As-Late-As-Possible (ALAP) scheduling approach, where operations are delayed and grouped towards the end of the circuit to minimize the time qubits spend in vulnerable states.

ALAP scheduling: This technique aims to reduce the impact of decoherence by performing operations as late as possible in the quantum circuit.

ECR gates: These are used throughout the diagrams, representing two-qubit operations that are susceptible to crosstalk and decoherence effects.

Software mitigation: The image references work by P. Murali et al. on software mitigation of crosstalk on noisy intermediate-scale quantum computers, presented at ASPLOS '20.

This information provides insight into the challenges of managing errors in quantum computing systems and some strategies for mitigating these issues through careful scheduling and operation design.

Many applications of near-term quantum computing involve layered circuits.

It describes a "Brickwork random circuit on a 1D chain".

The slide shows a diagram of this circuit, consisting of blue rectangles representing two-qubit random gates and purple lightning bolt symbols representing local noise.

Two main observations are listed:

The output of the circuit converges exponentially fast to the maximally mixed state.

Noise in the deep depth limit can be characterized by the global depolarizing noise.

A reference is cited: K. Tsubouchi et al., Universal cost bound of quantum error mitigation based on quantum estimation theory, arXiv: 2208.09385 (2023)

This slide continues the discussion on how noise propagates in quantum circuits, focusing on a specific case of error propagation and measurement.

The slide shows:

A diagram of a quantum circuit similar to the previous slide, with blue rectangles representing two-qubit gates and purple lightning bolts representing local noise.

A shaded pink area in the shape of a backwards light cone, stretching from the right side of the circuit diagram towards the left.

The text "U E" at the top left of the circuit, likely representing unitary operations and error.

"(O)" on the right side, possibly indicating an observable or output.

An explanation: "In the case of estimating expectation values, only errors in the backwards lightcone (shaded pink) contri35:48 Backwards we can shade the entire region and within this region those errors are going to contribute to the to the expectation values um this is pointed out in the paper reference blow and using this intuition we can argue that bute."

A citation: "M. Tran, K, Sharma, K, Temme, Locality and Error Mitigation of Quantum Circuits, arXiv:2303.06496, 2023"

The image discusses noise characterization in quantum state tomography, which is a process to learn an unknown quantum state ρ through experiments. Here's a summary of the key points:

Quantum state tomography aims to determine an unknown quantum state ρ experimentally.

The process involves a noise channel ε that transforms the input state ρ to an output state ρ' = ε(ρ).

For a single qubit, the density matrix ρ can be expanded as: ρ = [tr(ρ)I + tr(Xρ)X + tr(Yρ)Y + tr(Zρ)Z] / 2

The process requires:

Measuring the expectation value of the unknown state with all elements of a complete basis

Multiple copies of the unknown state

4^n - 1 different measurements for n qubits

This process is noted to be exponentially hard, making it an active area of research to find easier methods.

The image also shows a diagram representing the noise channel ε transforming ρ to ρ', and mentions that this is in a theoretical setup.

Is there any specific aspect of quantum state tomography or noise characterization you'd like to discuss further?

To measure the term tr(ρ)I in the context of quantum state tomography, we need to understand a few key points:

Meaning of tr(ρ)I:

tr(ρ) is the trace of the density matrix ρ.

I is the identity matrix.

For a valid quantum state, tr(ρ) should always equal 1.

Properties of the trace:

The trace of a density matrix is the sum of its diagonal elements.

It represents the total probability, which must sum to 1 for a properly normalized quantum state.

Measurement in practice:

Strictly speaking, you don't directly "measure" tr(ρ)I in an experiment.

Instead, this term is a consequence of the normalization of the quantum state.

Experimental verification:

While you don't measure tr(ρ)I directly, you can verify it indirectly through other measurements.

By performing a complete set of measurements to reconstruct the density matrix, you can calculate its trace.

If your measurements and reconstruction are accurate, the calculated trace should be very close to 1.

Importance in the expansion:

In the expansion ρ = [tr(ρ)I + tr(Xρ)X + tr(Yρ)Y + tr(Zρ)Z] / 2, the tr(ρ)I term ensures the proper normalization of the state.

Since tr(ρ) = 1 for a valid quantum state, this term effectively contributes I/2 to the expansion.

Relation to other measurements:

The other terms in the expansion (tr(Xρ)X, tr(Yρ)Y, tr(Zρ)Z) are what you actually measure in experiments.

These measurements, when properly normalized, implicitly confirm that tr(ρ) = 1.

In summary, while tr(ρ)I is not directly measured, it's a fundamental property of quantum states that is implicitly verified through the complete set of measurements performed in quantum state tomography. The actual experimental focus is on measuring the expectation values of the Pauli operators (X, Y, Z) for a qubit, which together with the normalization condition, fully characterize the quantum state.

This image discusses quantum process tomography, which is a method to characterize an unknown quantum process (represented by ε) through experiments. Here are the key points:

Objective: Learn an unknown quantum process ε from experiments.

Process representation: The process ε transforms an input state ρ to an output state ρ' = ε(ρ).

Input state preparation: Four input states need to be prepared: |n⟩ |m⟩ |+⟩ = (|n⟩ + |m⟩)/√2 |-⟩ = (|n⟩ + i|m⟩)/√2

Measurement: ε(ρ) is measured in 4^n complete basis states, where n is the number of qubits.

Complexity: ε has d^4 - d^2 independent real parameters, where d is the dimension of the Hilbert space.

Experimental requirement: 16^n experiments are needed, which is even more challenging than state tomography.

Research direction: Making quantum process tomography scalable is an active area of research.

Challenges: The process is exponentially hard as the number of qubits increases, making it difficult to scale to larger quantum systems.

This image explains the concept of randomized benchmarking (RB), a method used to characterize noise in quantum circuits. Here's a breakdown of the key elements:

Purpose: The aim is to quantify how much error occurs per gate in a quantum circuit.

Method: Single/Two-qubit randomized benchmarking

Circuit structure:

The circuit shows two qubits (q0 and q1) undergoing a series of operations.

Gates C1, C2, ..., Cm are applied sequentially.

Each Ci is a single or two-qubit random gate sampled from a finite Clifford gate set.

The final gate Cm+1 is chosen to make the total sequence equal to the identity operation.

Process:

Ci gates are randomly selected Clifford gates.

Cm+1 is specifically chosen to make the entire sequence equivalent to an identity operation.

After applying the sequence, the probability of returning to the ground state is measured.

The sequence length is varied, and the fidelity decay is fit to an exponential curve.

Analysis:

By varying the sequence length and fitting the fidelity decay to an exponential curve, the Error Per Clifford (EPC) can be reported.

Advantages:

This method allows for quantifying the average error per gate in a way that is less sensitive to state preparation and measurement errors compared to other techniques.

Limitation of RB: it does not capture coherent or crosstalk error due to randomization

This image explains how to extract the error rate per gate (EPG) in Randomized Benchmarking (RB). Here's a breakdown of the key points:

Fitting method: The survival probability is fit to an exponential function of the form Aα^m + B, where m is the number of Clifford gates.

Error calculation:

Average gate error: e = ((d-1)/d) * (1-α), where d is the dimension of the system

Average gate fidelity: F_g = (dF_p + 1) / (d + 1)

Average gate error per Clifford gate: e_g = 1 - F_g

Graph interpretation:

The x-axis represents the number of Clifford gates

The y-axis (P_1) represents the survival probability

Two datasets are shown: Reference (triangles) and Interleaved CNOT (circles)

The lines represent the exponential fit to these datasets

Results:

EPC (Error Per Clifford): 4.7 ± 0.1 × 10^-3

EPG (Error Per Gate): 2.3 ± 0.2 × 10^-3

Limitation: RB does not capture coherent or crosstalk errors

Source: The example is from a paper published in Physical Review Letters (Phys. Rev. Lett. 127.130501 (2021))

RB得到的

The image discusses layer fidelity in quantum computing, specifically the error per layer gate (EPLG) metric. This metric expands on randomized benchmarking and is used to assess the performance of quantum processors.

Key points from the image:

Layer fidelity measures the process fidelity of a layer of gates in a quantum circuit.

It involves N connected qubits and a set of connecting gates (entangling Clifford gates) on that device.

Layer structures are described as building blocks for many important quantum circuits.

The metric includes crosstalk effects.

It relates to other metrics to infer the error mitigation overhead.

The diagram shows a quantum circuit with 9 qubits (q0 to q9) and illustrates the concept of a layer of gates across these qubits.

This information is attributed to a paper by D. McKay et al., titled "Benchmarking Quantum Processor Performance at Scale," published on arXiv in 2023.

EPLG: can see the crosstalk effects.

The image shows a comparison of gate errors in quantum processors, specifically comparing the Eagle and Heron processors. The main components of the image are:

A graph showing a quantile plot of individual gate errors measured from the best 100 qubit chain. It compares "layered" (simultaneous direct RB) versus "isolated" (backend reported) gate errors for both Eagle and Heron processors.

An equation at the top: EPLG = 1 - LF^(1/n_2g)

Two diagrams showing the qubit layouts for the Eagle and Heron processors.

A caption stating "We can already see the effect of crosstalk error."

The presenter's comment about seeing "the effect of crosstalk error" refers to the differences between the "layered" and "isolated" measurements, where the layered measurements (which involve simultaneous operations) show higher error rates, possibly due to interference between qubits.