You can reduce noise in a quantum computer by employing several error mitigation and suppression techniques available through Qiskit Runtime. These techniques include:

• Dynamical decoupling: This technique inserts pulse sequences on idling qubits to approximately cancel out coherent errors caused by unwanted interactions between qubits. It is particularly useful for circuits with idle periods. You can enable it by setting estimator.options.dynamical_decoupling.enable = True and choose a sequence type like "XpXm".

• Pauli twirling: This method transforms any quantum channel into a Pauli channel by sandwiching chosen gates with randomly selected single-qubit Pauli gates. It helps mitigate coherent noise, which accumulates quadratically, by converting it into Pauli noise, which accumulates linearly. It is often applied to two-qubit gates. Enable it by setting estimator.options.twirling.enable_gates = True. You can also configure num_randomizations and shots_per_randomization.

• Twirled readout error extinction (TREX): TREX mitigates the effect of measurement errors when estimating Pauli observable expectation values. It uses twirled measurements, which involve randomly substituting measurement gates with a sequence of a Pauli X gate, a measurement, and a classical bit flip to diagonalize the readout-error transfer matrix. Enable it by setting estimator.options.resilience.measure_mitigation = True. Options for num_randomizations and shots_per_randomization are also available.

• Zero-noise extrapolation (ZNE): ZNE mitigates errors in estimating expectation values by executing the circuit multiple times at different noise rates (noise amplification) and then extrapolating the results to the zero-noise limit. Qiskit Runtime implements noise amplification via "digital gate folding". Enable it by setting estimator.options.resilience.zne_mitigation = True. You can specify noise_factors (e.g., (1,3,5)) and an extrapolator type (e.g., "exponential").

• Probabilistic Error Amplification (PEA): This is a more sophisticated approach to noise amplification, often used with ZNE. It performs preliminary experiments to reconstruct the noise model of each layer of entangling gates. This learned noise information is then used to accurately amplify noise by probabilistically injecting single-qubit noise proportional to the learned model at each entangling layer when executing circuits at different noise factors.

To reduce noise in a quantum computer, error mitigation techniques like Probabilistic Error Amplification (PEA) and Probabilistic Error Cancellation (PEC) can be used. These techniques utilize a noise learning component based on a Pauli-Lindblad noise model. While the error mitigation is typically managed during execution after submitting jobs, the NoiseLearner class in qiskit-ibm-runtime (version 0.27.1 or newer) allows you to explicitly obtain the results of these noise learning experiments.

Specifically, the NoiseLearner class performs experiments to characterize noise processes for one or more circuits. It has a run() method that executes these learning experiments, taking a list of circuits or a PUB as input, and returns a NoiseLearnerResult. This result contains the learned noise channels and metadata, which can then be stored locally and used as input for subsequent error mitigation experiments. The NoiseLearnerResult.data provides a list of LayerError objects, each containing a circuit, qubit labels, and the PauliLindbladError for the learned noise model, including its generators and error rates.

You can reduce noise in a quantum computer through error mitigation techniques, specifically Zero-Noise Extrapolation (ZNE) with Probabilistic Error Amplification (PEA).

Zero-Noise Extrapolation (ZNE)

ZNE removes the effects of an unknown noise during circuit execution by following these steps:

1. Amplify circuit noise for several noise factors (e.g., λ1, λ2,...).
2. Run every noise-amplified circuit to measure the expectation values (e.g., ⟨A(λ1)⟩,...).
3. Extrapolate back to the zero-noise limit ⟨A(0)⟩.

Noise Amplification for ZNE

There are three common ways to implement error amplification for ZNE:

• Pulse stretching: Scales pulse duration via calibration.
• Gate folding: Repeats gates in identity cycles.
• Probabilistic error amplification (PEA): Adds noise via sampling Pauli channels.

For utility-scale experiments, Probabilistic Error Amplification (PEA) is considered the most attractive method because it can be applied to any circuit that can run with a native noise factor and avoids the limitations of the other methods.

Learning the Noise Model for PEA

PEA requires learning the noise model, which can be done using a process similar to probabilistic error cancellation (PEC) in these steps:

1. Pauli twirl layers of two-qubit gates.
2. Repeat identity pairs of layers and learn the noise.
3. Derive a fidelity (error for each noise channel).

To reduce noise in a quantum computer, particularly depolarizing noise, the following methods can be applied:

1. Noise-estimation circuits: A method involves first estimating the rate of depolarizing noise using a dedicated noise-estimation circuit. After estimating the noise rate, the output of the target circuit is corrected based on this estimated rate.
2. Readout-error correction: This technique helps in mitigating errors that occur during the measurement or readout phase of quantum computations.
3. Randomized compiling: This method is used to transform coherent errors into stochastic errors, which can then be more easily mitigated.
4. Zero-noise extrapolation: This involves running quantum circuits at different noise levels and then extrapolating the results to the zero-noise limit to predict the error-free outcome.

To reduce noise in quantum computers, particularly in pre-fault-tolerant systems, one method is using Probabilistic Error Cancellation (PEC). This is an error-mitigation technique designed to obtain accurate, bias-free estimates of physical observables.

PEC works by effectively inverting well-characterized noise channels. A practical protocol involves learning and inverting a sparse noise model that can capture correlated noise and scale to large quantum devices. This approach has been demonstrated on superconducting quantum processors to address crosstalk errors.

To reduce the effects of noise in a quantum computer, particularly in noisy intermediate-scale quantum (NISQ) devices, one can employ quantum error mitigation methods. These methods are designed to suppress errors without requiring adaptive quantum operations, which are often not feasible with current NISQ technology. Instead, they involve:

1. Sampling Noisy Devices: Executing the quantum computation multiple times on available noisy devices to synthesize a series of distorted quantum states.
2. Classical Post-processing: Applying classical computational techniques to the measurement outcomes obtained from these multiple samples. This post-processing helps to suppress errors and estimate the true expectation value of an observable despite the inherent noise.

Examples of specific quantum error mitigation techniques mentioned include zero-error noise extrapolation, probabilistic error cancellation, and virtual distillation. The webpage also notes that quantum error correction is another approach, but it typically requires adaptive operations which are beyond the capabilities of many current NISQ devices.

To reduce noise in a quantum computer, particularly coherent noise which can be more damaging than incoherent noise, several methods can be employed:

• Pauli Conjugation: This method involves deterministically sandwiching the noise channel with a chosen pair of Pauli gates. The article proposes this technique to achieve higher logical fidelity compared to twirling, often with lower weights of conjugation gates.

• Pauli Twirling: This general solution involves using all possible Pauli gates to sandwich the noise channel and averaging the results, effectively turning coherent noise into a less damaging Pauli channel.

• Dynamical Decoupling: At the physical qubit level, coherent noise can be mitigated using this technique, although it has limitations due to imperfect control pulses and finite pulse durations.

• Improved Decoders: For local physical coherent noise at the logical level, its damage can be mitigated by using better decoders.

• Splitting Stabiliser Checks: Gate-level coherent errors in quantum error correction circuits can be mitigated by splitting the stabiliser check into two oppositely rotating halves, given certain architectural requirements for gates.

To reduce noise in a quantum computer, researchers at MIT developed a protocol called an "unbalanced echo." This method is analogous to noise-cancelling headphones.

The process involves characterizing how a specific source of noise, such as heat, affects nuclear quadrupole interactions within the system. By understanding this interaction, the team was able to use the same noise source to counteract nuclear-electron interactions, thereby extending the coherence times of nuclear-spin qubits from approximately 150 microseconds to as long as 3 milliseconds.

The researchers suggest that even greater improvements are possible by identifying and characterizing other sources of noise in the system, which could then also be canceled out.

Reducing noise in quantum computers is a significant challenge, and various techniques are being developed to address it. These approaches generally fall into two categories: error correction and error mitigation.

Quantum Error Correction (QEC)
QEC aims to achieve universal fault-tolerant quantum computation by encoding quantum states into multi-qubit entangled states. This method allows for the detection and correction of certain types of errors.

Quantum Error Mitigation (QEM)
QEM is an emerging field focused on improving the accuracy of near-term quantum computational tasks, often through data post-processing. It is considered a more feasible alternative to QEC for current quantum devices. Specific QEM protocols include:

• Zero-noise Richardson extrapolation: A technique to estimate noise-free values.
• Probabilistic error cancellation: Involves sampling circuits and taking their weighted average.
• Exploiting state-dependent bias: Uses invert-and-measure techniques to map the predicted state to the strongest one.
• Diverse ansatzs/models: Used to predict observables' noise-free values, distribution correction factors, or circuit noise metrics.
• Quantum subspace expansion: A method for error mitigation.
• Symmetry verification: Another approach to reduce errors.
• Learning-based techniques: Various methods that leverage machine learning to mitigate noise.

Measurement Error Mitigation (MEM)
MEM is a QEM protocol that models noise in a quantum circuit as a measurement noise matrix. This matrix is derived from probability distributions obtained by preparing and immediately measuring all possible basis input states.

Proposed Linear Algebraic Protocol
One approach, developed by the authors of the webpage, is a linear algebraic based protocol that efficiently models and mitigates the average behavior of noise. This method decomposes the average noise of a quantum circuit into State Preparation and Measurement (SPAM) error and average gate error. It is considered a passive technique, meaning only one noise characterization is made for the quantum device and then used to mitigate errors for any arbitrary circuit of specific depth on that device. The mitigation is performed using matrix algebra:

C_ideal = Q_m^(-1) C_noisy

Where (C_{ideal}) and (C_{noisy}) are the ideal and noisy outputs, respectively, and (Q_m) is the characterized noise matrix for circuits of a given depth (m).