Colin P. Williams

Automated Design of Quantum Circuits

In order to design a quantum circuit that performs a desired quantum computation, it is necessary to find a decomposition of the unitary matrix that represents that computation in terms of a sequence of quantum gate operations. To date, such designs have either been found by hand or by exhaustive enumeration of all possible circuit topologies. In this paper we propose an automated approach to quantum circuit design using search heuristics based on principles abstracted from evolutionary genetics, i.e. using a genetic programming algorithm adapted specially for this problem. We demonstrate the method on the task of discovering quantum circuit designs for quantum teleportation. We show that to find a given known circuit design (one which was hand-crafted by a human), the method considers roughly an order of magnitude fewer designs than naive enumeration. In addition, the method finds novel circuit designs superior to those previously known.

Computational synthesis of any n-qubit pure or mixed state

Future quantum information processing devices will require the use of exotic quantum states, such as specially crafted entangled states, to achieve certain desired computations on demand. Thus far, synthesis schemes for such states have been devised on a case-by-case basis using ad hoc techniques. In this paper we present a systematic method for finding a quantum circuit that can synthesize any pure or mixed n-qubit state. We then give examples of the use of our algorithm for finding synthesis pathways for especially exotic quantum states such as maximal mixed states. It is not known how to prepare general instances of such states by other means. Thus our quantum state synthesis algorithm should be of use not only in quantum information processing, but also in experimental quantum physics.

Quantum Chemistry with a Quantum Computer

In quantum chemistry, one is often interested in the static properties of a molecular quantum system, such its electronic structure, or its energy eigenvalues and eigenstates. In this chapter we describe the Abrams-Lloyd and Kitaev eigenvalue estimation algorithms. These provide efficient algorithms for determining the exact eigenvalue associated with a given eigenstate, a feat that is exponentially more difficult to do classically to the same precision.

Alternative Models of Quantum Computation

This chapter surveyed many superficially different models of quantum computation, which turned out to be polynomially equivalent to one another. This equivalency frees experimentalists to choose whichever model of quantum computation best fits the quantum physical phenomena they have at their disposal and over which they can exert control. We do not yet know which model of quantum computing will lead to the first genuinely scalable universal quantum computer.

Code Breaking with a Quantum Computer

Modern internet communications and electronic transactions rely heavily on the use of public key cryptosystems. Two famous examples are the RSA cryptosystem (RSA), and the elliptic curve cryptosystem (ECC). Such cryptosystems have the advantage that they do not require the sender and recipient of confidential messages to have met beforehand and exchanged secret key material. Instead, the person wishing to receive confidential messages creates a pair of matching public and private cryptographic keys, posts their public key for all to see, but keeps their private key secret. Anyone wishing to send the author of the public key a confidential message uses the posted public key to encrypt a message, and transmits the encrypted message via a potentially insecure classical communications channel. Upon receipt, the legitimate recipient uses his matching private key to unscramble the encrypted message.

Quantum Universality, Computability, & Complexity

In this chapter we surveyed issues of complexity, computability, and universality in the quantum and classical domains. Although there is no function a quantum computer can compute that a classical computer cannot also compute, given enough time and memory, there are computational tasks, such as generating true random numbers and teleporting information, that quantum computers can do but which classical ones cannot.

Quantum Error Correction

In this chapter we have looked at several approaches to dealing with errors in quantum computations. We found that it is not as easy to detect an error in a quantum computation as it is in a classical computation because errors may exist along a continuum of possibilities and our ability and we are not even allowed to read a corrupted state directly, because such direct observations would make matters worse rather than better.

Mathematics on a Quantum Computer

In this chapter we gave but a taste of these developments. The quantum counting algorithm is especially noteworthy since it combines ideas from both Grover’s algorithm and phase estimation. Moreover, quantum counting is practically useful as it can be used as a preliminary step in a quantum search when the number of solutions to the search problem is not known a priori. Hallgren’s algorithm for solving Pell’s equation is noteworthy because it represents a significant extension of Shor’s algorithm to the case of periodic functions having an irrational period. Quantum random walks are noteworthy because they have stimulated a great many new ideas for quantum algorithms and have also contributed to new physics understanding of transport phenomena in materials. Quantum algorithms that work on times series data and 2D images are possible, but these require that the data be encoded in a quantum state prior to the application of the quantum algorithm. This generally requires a computational cost that is proportional to the size of the data. Nevertheless, this could still be advantageous if subsequent quantum processing is exponentially faster than classical alternatives.

Solving NP-Complete Problems with a Quantum Computer

In the 1990’s a handful of computer scientists with a background in physics began looking at computational complexity from a fresh perspective. They wanted to know how the computational cost to solve an NP-Complete problem varied with the degree of constrainedness of the problem instances. They found that there is a critical value in constrainedness at which the difficulty of finding a solution rises steeply. Moreover, empirically, this region also coincides with an abrupt collapse in the probability of there being a valid solution. This has led to more physics-insight into analogies between the structure of NP-Complete problems and physical phase transitions.

Quantum Simulation with a Quantum Computer

Exact numerical simulations of quantum systems are intractable using classical computers for all but trivial systems. Approximate simulations can omit phenomena that are of critical importance to the system being simulated. And, last but not least, certain quantum phenomena are not intrinsically simulatable by any classical device unless we introduce artificial hidden variables. A fundamentally new approach to the simulation of quantum systems is needed.