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Methods for Quantum Computing

Final Report Summary - MQC (Methods for Quantum Computing)

Quantum computing (and, more broadly, quantum information science) combines quantum mechanics with computer science, by studying models of computers based on quantum mechanics. There has been a substantial progress on building a quantum computer over the last years. Individual operations of a quantum computer (known as quantum gates) can be now performed with precision 99.4%-99.92% and there is a number of efforts towards building larger scale quantum computers, from publicly funded research projects (such as the European Quantum Technologies Flagships) to large IT companies (such as IBM or Google) and startups and university spinoffs (such as Riggeti, IonQ or Alpine Quantum Computing).

The goal of the MQC project was to find applications for quantum computers when they are built (in other words: what software would we run on a quantum computer?). We aimed to construct new algorithms for quantum computers, to study the limits of what can be achieved with a quantum computer and to investigate how the ideas from quantum information can be used in other fields (including both quantum physics and conventional computer science).

Some highlights of our work are:
1. We developed new quantum algorithms for important search problems. Our first result is a quantum speedup for any classical search algorithm based on backtracking (a widely used classical search method that builds a tree consisting of partial solutions to the problem). Backtracking has applications from SAT solvers to optimization and circuit design and our quantum algorithm can be useful for all of those purposes. Our second result is a set of quantum algorithms for classically hard problems such as Travelling Salesman Problem (a well known optimization problem in which one has to plan the best route through a number of cities), providing the first provable quantum speedups for those problems.
2. We investigated the ultimate limits of quantum computers, by studying how large is the biggest possible advantage of quantum computers over classical computers. We found a computational task that is solvable by a quantum computer with just 1 access to input data but requires /sqrt{N} accesses for ordinary computers and showed that one cannot achieve substantially bigger advantage. This both shows the magnitude of quantum advantage that can be achieved and provides information for what types of tasks such advantage can be achieved.
3. We improved over the biggest known quantum speedup for the case if the quantum algorithm must output a definite answer on all input data (i.e. if the algorithm has to compute a total function). In this case, the biggest speedups are substantially smaller and it was conjectured that the quadratic speedup achieved by Grover's quantum search algorithm is the best possible. We disproved that, discovering a total function for which the quantum complexity is roughly the 4th root of the classical complexity. This direction has also lead to a solution of a 30-year old open problem about the biggest gap between two classical models of computation, randomized and deterministic query algorithms.
4. We investigated the security of cryptographic schemes against a quantum attacker. It is well known that quantum computers can solve some of the difficult problems which serve as a basis for secure data transmission. We showed a new source of insecurity: even if a scheme is based on a problem that is hard for quantum computers, it can be insecure because of its security proof using constructions that fail in the quantum case.
5. We studied the complexity of classical (non-quantum) algorithms for matrix multiplication (which is among the most important computational tasks). We showed a limit for a class of algorithms for matrix multiplication that includes all the best of algorithms of the last 25 years (Coppersmith-Winograd algorithm and all of its improvements). This shows that, for a substantial advance in this area, important new ideas are necessary. While the result involves no quantum computers, it is of an algebraic nature, similar to much of the research in quantum computing.