Quantum Computing with Very Noisy Devices

There are quantum algorithms that can efficiently simulate quantum physics, factor large numbers and estimate integrals. As a result, quantum computers can solve otherwise intractable computational problems. One of the main problems of experimental quantum computing is to preserve fragile quantum states in the presence of errors. It is known that if the needed elementary operations (gates) can be implemented with error probabilities below a threshold, then it is possible to efficiently quantum compute with arbitrary accuracy. Here we give evidence that for independent errors the theoretical threshold is well above 3%, which is a significant improvement over that of earlier calculations. However, the resources required at such high error probabilities are excessive. Fortunately, they decrease rapidly with decreasing error probabilities. If we had quantum resources comparable to the considerable resources available in today’s digital computers, we could implement non-trivial quantum algorithms at error probabilities as high as 1% per gate. Research in quantum computing is motivated by the great increase in computational power offered by quantum computers.1–3 There is a large and still growing number of experimental efforts whose ultimate goal is to demonstrate scalable quantum computing. Scalable quantum computing requires that arbitrarily large computations can be efficiently implemented with little error in the output. Criteria that need to be satisfied by devices used for scalable quantum computing have been specified by DiVincenzo.4 One of the criteria is that the level of noise affecting the physical gates is sufficiently low. The type of noise affecting the gates in a given implementation is called the “error model”. A scheme for scalable quantum computing in the presence of noise is called a “fault-tolerant architecture”. In view of the criterion above, studies of scalable quantum computing involve constructing fault-tolerant architectures and providing answers to questions such as the following: Q1: Is scalable quantum computing possible for error model E? Q2: Can fault-tolerant architecture A be used for scalable quantum computing with error model E? Q3: What resources are required to implement quantum computation C using fault-tolerant architecture A with error model E? To obtain broadly applicable results, fault-tolerant architectures are constructed for generic error models. Here, the error model is parametrized by an error probability per gate (or simply error per gate, EPG), where the errors are unbiased and independent. The fundamental theorem of scalable quantum computing is the threshold theorem and answers question Q1 as follows: If