### Introductions

#### In 1982, Richard Feynman gave a lecture at the MIT Computer Science and Artificial Intelligence Laboratory. He proposed using quantum mechanical phenomena to perform calculations that would be impractical or impossible using classical computers. He proposed that the best way to simulate a quantum mechanical system would be in a quantum computer.

#### Little did he know that this idea would catapult into an entire branch of science that is studied and worked on extensively today with new growth every day. This field of Quantum Computing is growing rapidly but is held back by many errors like quantum noise, decoherence, crosstalk, gate control inaccuracies, etc.

#### There are various reasons why errors can occur in quantum computers. One of the main reasons is the inherent fragility of qubits, which are the building blocks of quantum computers. Qubits are extremely sensitive to their environment and can be easily disturbed by factors such as electromagnetic noise, temperature fluctuations, and vibrations.

#### Another reason is that quantum algorithms are highly complex and require a large number of operations, which increases the likelihood of errors occurring. In addition, the process of measuring a qubit can cause it to collapse from a quantum state to a classical state, which can introduce errors into the computation.

#### Furthermore, the current state of quantum technology is still in its early stages, and the development of robust and fault-tolerant quantum hardware and software is still ongoing. This means that quantum computers are currently more prone to errors than classical computers, and error correction techniques are required to mitigate these errors.

#### The best way to understand the effect of these noises and errors is through an example. The following circuit comprise of a Hadamard Gate, a Control Not Gate and a Not gate applied to two different qubits as shown below.

The circuit simulated on Qiskit

#### When this circuit is simulated on the Qasm simulator the for 500 shots the results obtained is as follows.

#### This result is much more consistent with what the mathematical analysis of circuit as the state that we would get after application of the circuit would be of the form.

## |ÏˆâŸ© =(|01âŸ©+|10âŸ©)/âˆš2

#### The state has equal probability to be in either state and thus the result is much more consistent with the simulator but when this same circuit is run on Quantum Computer the result is much more different as shown in the fig below.

#### The above that even though we get state and most of the time. We still get a small number of results for the other states as well. This is due to various errors such as the noise discussed above.

#### Now that you are familiar with errors in a QC it is time to understand how some methods are proposed to handle these errors. Their two specific ways to handle these errors. Error correction is a technique that can be used to reduce the number of errors in quantum computations. There are a number of different error correction techniques, but they all work by encoding the qubits in a way that makes them more resistant to noise.

#### One of the most common error correction techniques is the Shor code. The Shor code encodes a single qubit into nine physical qubits. This allows the code to correct for both bit flips and phase flips, which are the two most common types of errors that can occur in quantum computers.

#### Another popular error correction technique is the surface code. The surface code is a more complex code than the Shor code, but it can correct for a wider range of errors. The surface code is also more scalable than the Shor code, which means that it can be used to build larger quantum computers.

### Error Correction:

#### Error correction is a critical technology for the development of quantum computers. Without error correction, it would be impossible to perform accurate quantum computations. However, error correction is also a challenging problem, and there is still much research to be done in this area.

#### In addition to the Shor code and the surface code, there are a number of other error correction techniques that are being developed. These techniques include the Toric code, the color code, and the droplet code.

#### The development of new error correction techniques is an active area of research. As quantum computers become more powerful, the need for more sophisticated error correction techniques will become even more pressing.

#### Still there are a few challenges with error correction codes which are, Error rates. The error rate of a quantum computer is the probability that an error will occur during a quantum computation. The lower the error rate, the more accurate the results of the computation will be. However, it is very difficult to achieve low error rates in quantum computers. Complexity. Quantum error correction codes can be very complex. This makes them difficult to implement in hardware, and it also makes them difficult to analyze mathematically. Scalability. Quantum error correction codes need to be scalable in order to be used in large-scale quantum computers. This means that they need to be able to correct for errors in a large number of qubits. Due to these issues error mitigation strategies are proposed.

### Error Mitigation:

#### These strategies aim to reduce the error instead of removing it completely. QEM is important because it can help to improve the accuracy of quantum algorithms, even when the underlying hardware is noisy. There are a number of different QEM strategies, but they all work by exploiting the statistical properties of noise. For example, some QEM strategies involve repeating a quantum computation multiple times and then averaging the results. This helps to reduce the impact of individual errors, because the errors are likely to cancel each other out.

**Repeating measurements:** This is the simplest QEM strategy. It involves repeating a quantum measurement multiple times and then averaging the results. This helps to reduce the impact of individual measurement errors. This is also called Symmetrization and Aggregation.

**Probabilistic error cancellation: **This strategy involves using classical computers to analyze the results of quantum measurements and identify errors. This information can then be used to correct the errors or to improve the accuracy of the results.

#### Error extrapolation: This strategy involves running a quantum computation at different noise levels and then extrapolating the results to zero noise. This helps to identify the errors that are caused by noise and to correct for them.

**Error suppression: **This strategy involves using classical computers to control the quantum hardware in a way that reduces the impact of noise. For example, error suppression techniques can be used to optimize the timing of quantum gates or to apply compensating pulses to the qubits.

### What is Symmetrization?

#### Symmetrization is a technique used in error mitigation on quantum computers. It involves running a quantum circuit multiple times with random variations in the input parameters, and then taking the average of the results to obtain a more accurate estimate of the true output of the circuit.

#### The basic idea behind Symmetrization is that by introducing random variations in the input parameters, the effects of noise and errors on the quantum circuit can be distributed more evenly, thereby reducing their impact on the final result. By averaging the results over many trials, the random variations cancel out, and the true output of the circuit can be recovered with greater accuracy.

#### In a quantum computer, Symmetrization is typically applied to a quantum circuit by running the circuit multiple times with different input parameters. The input parameters can be randomly varied, or they can be systematically varied in a way that preserves certain symmetries or invariances of the circuit.

#### For example, suppose we have a quantum circuit that takes as input a set of qubits and performs a sequence of gates and measurements to produce an output. To apply Symmetrization, we would first choose a set of input parameters, such as the angles of the gates in the circuit. We would then run the circuit multiple times with these input parameters, but with small random variations introduced in each trial.

#### After running the circuit multiple times, we would collect the results of the measurements for each trial and compute the average value of the measurements. This average value would give us a more accurate estimate of the true output of the circuit, as it would help to mitigate the effects of noise and errors on the measurements. To further improve the accuracy of the results, we could repeat this process with different sets of input parameters, or we could systematically vary the input parameters in a way that preserves certain symmetries or invariances of the circuit. For example, we could vary the input parameters in a way that preserves rotational or translational symmetry of the circuit.

#### In the present landscape for mitigating errors and improving the accuracy of quantum computations, Symmetrization is considered a powerful technique, and it is widely used in various applications of quantum computing, such as quantum chemistry and optimization.

### How can Aggregation be used alongside Symmetrization in a Quantum Computer?

#### Aggregation refers to the process of combining multiple quantum systems into a single larger system. This can be done in a variety of ways, such as by bringing two or more particles together to form a composite system, or by combining the states of multiple qubits to form a larger quantum register.

#### In the context of quantum mechanics aggregation turns out to be an important concept, as it allows us to describe the behavior of complex quantum systems in terms of simpler subsystems. By studying the properties of these subsystems and how they interact with one another, we can gain insights into the behavior of the larger system as a whole.

#### As a result of the above characteristic, aggregation of qubits turns out to be a crucial aspect in quantum computing as it allows for the creation of larger and more powerful quantum systems. The more qubits that are entangled, the more complex calculations and simulations can be performed, potentially enabling breakthroughs in fields such as cryptography, materials science, and drug discovery.

#### In the context of error mitigation, it refers to the process of grouping together a set of quantum circuits or measurements to reduce the impact of noise and errors on the overall result. This technique takes advantage of the fact that certain types of errors, such as random bit flips, are likely to occur independently in different parts of the circuit or measurement process. By aggregating these parts, the errors can cancel out, resulting in a more accurate overall result.

#### In quantum computing, one approach to perform aggregation is to use multiple runs of the same quantum computation or measurement, and then take an average or median of the results. This can help to reduce the impact of noise and errors in the measurement or computation process, and improve the accuracy of the final result. This technique is commonly used in quantum annealing or optimization algorithms, where multiple runs are performed to find the lowest-energy configuration of a system.

#### It can further be used for techniques such as error correction codes or error detection codes. These involve encoding the quantum information in a redundant way to detect and correct errors that may occur during computation. These codes can be designed to aggregate or group together qubits in a way that maximizes the effectiveness of error correction or detection.

### What is plurality voting and how could it be used in Aggregation?

#### The candidate that obtains the most votes is considered the victor under a plurality voting system, in which each voter picks their favorite candidate. Voters may normally choose just one candidate from a list of candidates in a plurality vote, and the candidate who receives the most votes wins, even if they do not get an absolute majority of the vote (more than 50%).

#### The aggregate phase is another instance when this method can be utilized. Aggregation via plurality voting is a technique for combining the outcomes of many quantum circuits, where each circuit's output is a binary measurement outcome (for example, either 0 or 1). In this method, the circuit findings are combined by aggregating the measurement results for each qubit over all circuits.

**Consider the following scenario: **

#### We measure each qubit in each of three quantum circuits, each of which has two qubits. The three circuits' measurement results can resemble this:

#### Circuit 1: 0 1

#### Circuit 2: 1 1

#### Circuit 3: 0 1

#### We would count the number of 0s and 1s for each qubit and take the majority vote for each qubit in order to combine these findings using plurality voting. Qubit 1 has two 0s and one 1 in this instance, making 0 the preferred outcome. With two 1s and a 0 in Qubit 2, the majority decision is a 1. As a result, the two-qubit system's aggregate measurement result is (0, 1).

#### A quick and effective way to combine measurement data from various quantum circuits is to use plurality voting. It may not be the best option in some circumstances since it does not account for correlations between the measurement results for various qubits or circuits. These correlations can be taken into consideration by other aggregation techniques like maximum likelihood estimation and Bayesian inference, which in some circumstances may yield more precise findings.

**Author - Kartike Pushkarna**

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