Routes towards universality in practical quantum computation
Formally, universality refers to the property of a set of quantum gates; they are universal if, when composed into circuits, they are capable of implementing any (arbitrarily good approximation of a) unitary transformation on qubits.
This formal property is possessed by almost any collection of gates [D94]; but the effective achievement of a practical universal quantum computer is currently beyond our grasp, and drives a wide-ranging collection of research thrusts.
We have been endeavoring for many years to work closely with experimentalists to make superconducting devices which can implement quantum gates sufficiently close to unitary (i.e., sufficiently free of decoherence) that they can be used in universal quantum computation. These efforts have very gradually met with success [KKMR*06,SKDR*10]. Many concepts of effective control have been developed in the course of this work which remain to be applied further as new technical improvements emerge in the devices: There is "the portal" (the controlled switching between adiabatic and non-adiabatic evolution [KRKM*05]), "parking" (the controlled transfer of quantum information from qubits to superconducting cavities, and alternative way of exploiting circuit quantum electrodynamics [ABDP*08]), linear optics concepts and the role of pieces of the Hamiltonian beyond the rotating wave approximation [CBKD10]. Through all of this it has been important to be able to derive effective Hamiltonians for these systems at various levels of detail [BKD03].
New projects are certain to develop in this area as the subject continues to progress. An active collaboration will be maintained with the IBM experimental superconducting research group in Yorktown Heights, New York. It is intended that this project will couple strongly to investigations of fault tolerant architectures, described below.
Joint theory-experiment projects are planned. We will investigate varying the nuclear spin physics by assessing new physical systems (pure Ge quantum wells, C-based structures, II-VI materials). New coupling schemes for multi-qubit experiments are vital, and we will pursue the possibilities afforded by superconducting quantum resonators.
The notion of "effective" universal quantum computation is very much driven by the notions of error correction. Defeating the effects of decoherence is one of the essential requirements of quantum computation [D00]. The known effective quantum error correction codes involve specific one- and two-qubit operations; we now know that they can be effective in a planar geometry, in which only short-distance couplings between qubits in two dimensions are needed.
New research will focus on whether the standard scheme, involving the Bravyi-Kitaev surface code [BK98] , can be improved upon. This scheme requires four-qubit local measurements, typically implemented using an ancilla and a set of CNOT operations. Topological subsystem codes which involve non-commuting operators [SBT10] have possible advantages over the Bravyi-Kitaev surface code, but remain to be further explored.
The essence of quantum error correction is the measurement of sets of commuting operators. It has always been assumed that these multi-qubit operators would be measured by the operation of a quantum circuit followed by single-qubit measurements. But it is also possible to consider physical settings in which the required multi-qubit non-demolition measurements can be implemented directly. We will look for such approaches within the circuit quantum electrodynamics paradigm.
Another strand of research on quantum error correction pertains to the notion of 'self-correcting' quantum computers which are not actively stabilized by quantum measurements but by the presence of macroscopic energy barriers preventing error excitations from accumulating. Self-correction has been shown to be possible for quantum systems in four spatial dimensions [DKLP01] and strong evidence exists for its impossibility in two dimensions [BT00]. The situation in three dimensions is not well understood.
Solid state qubits will need a multitude of precise, efficient generators and detectors of classical signals, both analog and digital, available at low temperatures, for the initialization, clocking, and feedback measurements of the quantum computer. While this is not currently considered to be in the domain of theoretical physics, we will be working to understand the feasibility of proposed [M11] low temperature digital devices in which the superconducting flux quantum embodies the bit.
Kitaev pointed out that Majorana modes have unique possibilities for carrying quantum information reliably [K00]. We see important directions both for fundamental and practical studies. The "unpaired Majorana fermion" (also referred to as "half a fermion") can be the basis of new types of quantum error correction codes [BLT10]. There are possibilities for achieving Majorana modes in superconductors, and in hybrid structures involving topological insulators. We intend to follow up recent proposals of Kane and co-workers [FK07, JKP10] to achieve such modes, and quantum gate operations involving them. This will involve further work on coherent superconducting structures, and their functioning together with topological insulators. We will help to plan the basic experiments for seeing the Majorana mode spectroscopically in superconductor-topological insulator heterostructures.
A very important paper [KTR05] took up the question of whether the Hamiltonian introduced by Feynman, which is a paradigm of a system whose ground state energy is hard to compute, is "realistic", in the sense that it can be realized by two-body operations. (The Feynman Hamiltonian as written is realized directly by five-spin operations.) [KTR05] realized that to answer this question positively, it was necessary to view the five-spin Hamiltonian as the effective Hamiltonian for the low-energy excitiations of a system in a larger-dimensional space. The enlargement of the Hilbert space is achieved by "gadgets", and the relation between the Hamiltonians is achieved by many-body perturbation theory. We have made this application of perturbation theory more systematic, and have found that the formalism of Schrieffer and Wolff fits the needs of this application very well. We are still working on filling out the formal structure of this theory, i.e., in proving convergence properties, and showing linked-cluster properties.
As we anticipate a future world in which a quantum computer could similate the dynamics of any local Hamiltonian, one may ask what are those properties particular to local Hamiltonians which distinguish them from 2n x 2n Hermitian matrices. Complexity theory asks about the hardness of the computational problems pertaining to such Hamiltonians. By asking and answering the computational question, it sheds light on structural properties of these Hamiltonians. These structural properties may depend for example on geometric locality, presence of a gap, properties of the excitation spectrum, or whether the Hamiltonian avoids the sign problem (i.e. is stoquastic [BT08]).