Simplifying quantum logic using higher-dimensional Hilbert spaces

Abstract

Quantum computation promises to solve fundamental, yet otherwise intractable, problems across a range of active fields of research. Recently, universal quantum logic-gate sets-the elemental building blocks for a quantum computer-have been demonstrated in several physical architectures. A serious obstacle to a full-scale implementation is the large number of these gates required to build even small quantum circuits. Here, we present and demonstrate a general technique that harnesses multi-level information carriers to significantly reduce this number, enabling the construction of key quantum circuits with existing technology. We present implementations of two key quantum circuits: the three-qubit Toffoli gate and the general two-qubit controlled-unitary gate. Although our experiment is carried out in a photonic architecture, the technique is independent of the particular physical encoding of quantum information, and has the potential for wider application.T he realization of a full-scale quantum computer presents one of the most challenging problems facing modern science. Even implementing small-scale quantum algorithms requires a high level of control over multiple quantum systems. Recently, much progress has been made with demonstrations of universal quantum gate sets in a number of physical architectures including ion traps 1,2 , linear optics 3-6 , superconductors 7,8 and atoms 9,10 . In theory, these gates can now be put together to implement any quantum circuit and build a scalable quantum computer. In practice, there are many significant obstacles that will require both theoretical and technological developments to overcome. One is the sheer number of elemental gates required to build quantum logic circuits.Most approaches to quantum computing use qubits-the quantum version of bits. A qubit is a two-level quantum system that can be represented mathematically by a vector in a two-dimensional Hilbert space. Realizing qubits typically requires enforcing a twolevel structure on systems that are naturally far more complex and which have many readily accessible degrees of freedom, such as atoms, ions or photons. Here, we show how harnessing these extra levels during computation significantly reduces the number of elemental gates required to build key quantum circuits. Because the technique is independent of the physical encoding of quantum information and the way in which the elemental gates are themselves constructed, it has the potential to be used in conjunction with existing gate technology in a wide variety of architectures. Our technique extends a recent proposal 11 , and we use it to demonstrate two key quantum logic circuits: the Toffoli and controlled-unitary 12 gates. We first outline the technique in a general context, then present an experimental realization in a linear optic architecture: without our resource-saving technique, linear optic implementations of these gates are infeasible with current technology. Simplifying the Toffoli gateOne of the most important quantum logic gates is the Toffoli 1...

Heralded processes on continuous-variable spaces as quantum maps

“…(8). It can be checked that using this formula as the definition, and using the relation between E n,m l,k andÊ i , one arrives at the original definition.…”

“…In the context of optical quantum information, conditional evolution has found several applications for simulating strong nonlinearities at the few-photon level. This approach has allowed researchers to build two-qubit [3] and three-qubit quantum logic gates [8] and to generate quantum states with non-Gaussian Wigner functions [9][10][11][12][13]. Such an evolution is able to induce non-Gaussian transformations effective in overcoming existing no-go theorems valid for purely Gaussian resources [14][15][16].…”

Observation of Four-Photon Orbital Angular Momentum Entanglement

“…For photons, the spatial degrees of freedom enable high-dimensional single particle spaces, which can be discretized in the photon orbital angular momentum (OAM). This enables implementation of novel quantum information protocols [3][4][5], and the study of fundamentally new quantum states [6,7]. To date, only two such multidimensional particles have been entangled [8,9] albeit with ever increasing dimensionality [10][11][12]; only in continuous variables, a first study goes beyond this [13].…”

Hidden two-qubit dynamics of a four-level Josephson circuit

“…Many physical systems commonly used as qubits, such as trapped ions 4,5 and superconducting circuits 6,7 , are in reality multi-level systems whose dynamics are manually limited by researchers to two levels to preserve simplicity and fidelity. Such systems can easily be employed as multi-level qudits by accessing additional levels, allowing a significant reduction of the resources and gates required for a variety of quantum information applications [8][9][10] , yet also introducing a complex multi-level problem. The usefulness and popularity of multi-level qudits are limited by a dearth of theoretical methods for understanding and controlling their dynamics.…”