The earliest notion of a quantum computer was conceptualized by Feynman as a “powerful computer which can solve complex problems of quantum physics and chemistry more efficiently than a classical computer”. The idea is based on the fundamental difference between quantum and classical information. More accurately, that the information carried by a quantum system that is sufficiently well isolated from its surroundings, has intrinsic features which are not shared by classical information such as randomness, uncertainty and entanglement. The vision about possible applications of such a computer got more clear with the advent of quantum algorithms like the Deutsch–Jozsa algorithm and Shor’s algorithm that provided a framework to solve large factoring problems which are hard for classical computers. It was later realized that such computing hardware is practically feasible to construct and can be scalable to large processing systems. In this historical and pedagogical overview, Preskill summarizes and comments on the past 40 years of quantum computing research and development, and what’s next for the field.

The general features of a quantum computer model, based on the quantum circuit model, are: i) a scalable number of qubits, ii) preparation of standard initial states, iii) universal set of quantum gates acting on the current state to result in desired final state, iv) a quantum circuit and v) a readout in the standard basis. A significant advantage of such a model is that the runtime for simulating an n-qubit quantum system using a classical computer rises exponentially with n, while the runtime for simulating the system on a quantum computer scales as a power of n, i.e. polynomially. Such hardware has been steadily improving throughout the current NISQ era. An era that has showcased a Google-constructed programmable quantum computer called Sycamore with 53 working qubits which executed up to 20 layers of two-qubit gates, and then measured all the qubits millions of times in just a few minutes, extracting a statistically useful signal. Furthermore, with the evolution of quantum error correction in conjunction to the novel fault-tolerant methods for executing reliable quantum computation while using noisy hardware, the NISQ systems are undoubtedly evolving to systems capable of running more and more sophisticated quantum algorithms.

Another significant comparison indicating the advancements of quantum computing is between analog and digital quantum simulation. So far, analog quantum simulators not only possess better capabilities as compared to their classical counterparts, but they are also compatible with most of the existing experimental platforms like trapped ions, superconducting circuits and trapped neutral atoms and molecules. However, they are limited by imperfect control due to the large amounts of noise. On the other hand, despite being more costly, digital quantum simulations offer greater flexibility in processing the desired Hamiltonians and preparation of the standard initial states. At the moment, a combination of both analog and digital simulation seems to be producing the best results, although the anticipation for the future is that digital simulations will be heavily used.

The progress achieved so far to make the transition from NISQ to Fault-Tolerant Quantum Computing has been driven by advances in qubit design, control technology, fabrication methods, and materials. A critical improvement that will highly affect the performance of the quantum computer, is the improvement of physical gate error rates by several orders of magnitude. For example, currently the error rate is typically 1% for entangling two-qubit gates. Alongside these developments, quantum algorithms should also continue to be explored, so that new ways of achieving quantum speedups in various problems like optimization can become feasible. It is theorized that perhaps a quadratic speedup can be achieved for optimization related problems. While researchers are already exploring challenging problems with NISQ devices like investigating properties of highly entangled many-body quantum systems, scalability of fault-tolerant quantum computers is another significant area to focus on, in order to make more powerful and efficient systems.

An interesting outlook in the paper is that quantum computing research does not only benefit the hardware and application aspects, but also the fundamental physics understanding of many-body physics, quantum information and entanglement, and even supports developments in unification of quantum models of gravity. The author concludes that from now on, “quantum computer

science and quantum physical science will advance together, hand in hand”.