Quantum Computing Advances: Error Mitigation Techniques Enhance Noisy Quantum States

Quantum error mitigated classical shadows are a concept in quantum computing that allows for learning many properties of a quantum state with few measurements. This is particularly useful for early quantum computers that can only prepare noisy quantum states. Techniques such as probabilistic error cancellation (PEC), zero noise extrapolation (ZNE), and symmetry verification (SV) have been developed to mitigate errors. PEC shadows are an unbiased estimator for the ideal quantum state. The future of quantum computing looks promising, with the potential to solve computational tasks that are currently impossible or extremely difficult with classical computers.

What are Quantum Error Mitigated Classical Shadows?

Quantum error mitigated classical shadows are a concept in quantum computing that allows us to learn many properties of a quantum state with very few measurements. This concept is particularly relevant for near-term and early fault-tolerant quantum computers, which will only be able to prepare noisy quantum states. The challenge lies in efficiently learning properties of an ideal, noise-free state.

Error mitigation techniques such as probabilistic error cancellation (PEC), zero noise extrapolation (ZNE), and symmetry verification (SV) have been developed for mitigating errors in single expected value measurements. These techniques have been generalized for mitigating errors in classical shadows. PEC is considered the most natural candidate for this, and a thorough theoretical framework for PEC-shadows has been developed.

PEC shadows are an unbiased estimator for the ideal quantum state. The sample complexity for simultaneously predicting many linear properties of the ideal state is identical to that of the conventional shadows approach, up to a multiplicative factor which is the sample overhead due to error mitigation. This overhead does not depend directly on the number of qubits but rather grows exponentially with the number of noisy gates.

How are Quantum Computers Developing?

Quantum computers are developing rapidly and can already perform certain demonstration tasks that are impossible or very difficult with even the largest supercomputers. However, it is still to be seen whether the technology can achieve true practical quantum advantage, i.e., the point when these machines can solve an otherwise impossible computational task that is of value to industry or to researchers in other fields such as quantum field theory, quantum gravity, drug development, and materials science.

Quantum computers are highly vulnerable to noise and while quantum error correction provides a comprehensive solution, implementing it poses an extreme engineering challenge. It is generally expected that some form of early practical quantum advantage, just beyond the reach of classical computing, could be achieved even with noisy quantum computers. This prospect has motivated the development of a broad range of quantum error mitigation protocols.

What are the Challenges and Solutions in Quantum Computing?

One major challenge in quantum computing is that near-term quantum algorithms typically require an extreme number of circuit repetitions in order to suppress shot noise. Classical shadows were introduced relatively recently and represent another promising angle in achieving practical quantum advantage. The approach allows one to extract many properties of a quantum state without having to repeat the measurement many times. This is achieved by performing measurements in randomized bases.

The measurement outcomes, as bitstrings along with the indexes of the measurement bases, form a classical shadow which is an efficient classical representation of the entire quantum state. Shadows have become an entire subfield and various promising applications have been proposed that greatly benefit from the rich information one can access via shadows.

How is Quantum Error Mitigation Connected with Classical Shadows?

The focus of the present work is to amalgamate quantum error mitigation techniques with classical shadows. Prior works have considered fruitful connections between quantum error mitigation and classical shadows. For instance, some use classical shadows obtained from a noisy quantum state to perform purification-based error mitigation offline with access only to a single copy of the state but at an exponential complexity in the number of qubits.

However, the present work addresses a distinct problem where the state is generated by a noisy quantum circuit and the aim is to mitigate the impact of errors induced by the noisy quantum gates. The focus is thus to extract properties of an ideal state which would be generated by a noise-free quantum computer. This approach is a generalization of quantum error mitigation techniques.

What is the Future of Quantum Computing?

The broad set of tools introduced in this work may be instrumental in exploiting near-term and early fault-tolerant quantum computers. Detailed numerical simulations demonstrate a range of practical applications of quantum computers that will significantly benefit from these techniques.

As quantum computers continue to develop and improve, the techniques and concepts discussed in this work will become increasingly relevant. The ability to mitigate errors and extract valuable information from noisy quantum states will be crucial in achieving practical quantum advantage. The future of quantum computing looks promising, with the potential to solve computational tasks that are currently impossible or extremely difficult with classical computers.