ARTICLES PUBLISHED ONLINE: 7 DECEMBER 2008 DOI: 10.1038/NPHYS1150 Simplifying quantum logic using higher-dimensional Hilbert spaces Benjamin P. Lanyon1*, Marco Barbieri1, Marcelo P. Almeida1, Thomas Jennewein1,2, Timothy C. Ralph1, Kevin J. Resch1,3, Geoff J. Pryde1,4, Jeremy L. O’Brien1,5, Alexei Gilchrist1,6 and Andrew G. White1 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. he realization of a full-scale quantum computer presents one Simplifying the Toffoli gate of the most challenging problems facing modern science. One of the most important quantum logic gates is the Toffoli12— TEven implementing small-scale quantum algorithms requires a three-qubit entangling gate that flips the logical state of the a high level of control over multiple quantum systems. Recently, `target' qubit conditional on the logical state of the two `control' much progress has been made with demonstrations of universal qubits. Famously, these gates enable universal reversible classical quantum gate sets in a number of physical architectures including computation, and have a central role in quantum error correction13 ion traps1,2, linear optics3–6, superconductors7,8 and atoms9,10. In and fault tolerance14. Furthermore, the combination of the Toffoli theory, these gates can now be put together to implement any and the one-qubit Hadamard offers a simple universal quantum quantum circuit and build a scalable quantum computer. In gate set15. The simplest known decomposition of a Toffoli when practice, there are many significant obstacles that will require both restricted to operating on qubits throughout the calculation is a theoretical and technological developments to overcome. One is circuit that requires five two-qubit gates12. If we further restrict the sheer number of elemental gates required to build quantum ourselves to controlled-z (or cnot) gates, this number climbs to logic circuits. six12 (Fig. 1a). A decomposition that requires only three two-qubit Most approaches to quantum computing use qubits—the gates11 is shown in Fig. 1b. The increased efficiency is achieved by quantum version of bits. A qubit is a two-level quantum system that harnessing a third level of the target information carrier—the target can be represented mathematically by a vector in a two-dimensional is actually a qutrit with logical states j0i, j1i and j2i. Hilbert space. Realizing qubits typically requires enforcing a two- At the input and output of the circuit, information is encoded level structure on systems that are naturally far more complex only in the bottom two (qubit) levels of the target. The action of the and which have many readily accessible degrees of freedom, first Xa gate is to move information from the logical j0i state of the such as atoms, ions or photons. Here, we show how harnessing target into the third level (j2i), which then bypasses the subsequent these extra levels during computation significantly reduces the two-qubit gates. The final Xa gate then coherently brings this number of elemental gates required to build key quantum circuits. information back into the j0i state, reconstructing the logical qubit. Because the technique is independent of the physical encoding By temporarily storing part of the information in this third level, of quantum information and the way in which the elemental we are effectively removing it from the calculation—enabling the gates are themselves constructed, it has the potential to be used subsequent two-qubit gates to operate on a subspace of the target. in conjunction with existing gate technology in a wide variety of This enables an implementation of the Toffoli with a significantly architectures. Our technique extends a recent proposal11, and we reduced number of gates. Note that only standard two-qubit gates use it to demonstrate two key quantum logic circuits: the Toffoli are necessary, with the extra requirement that they act only trivially and controlled-unitary12 gates. We first outline the technique in on (that is, apply the identity to) level j2i of the qutrit. As such, it is a general context, then present an experimental realization in a not necessary to develop a universal set of gates for qutrits. linear optic architecture: without our resource-saving technique, This technique can be readily generalized to implement linear optic implementations of these gates are infeasible with higher-order n-control-qubit Toffoli gates (nt) by harnessing a current technology. single (nC1)-level information carrier during computation and 1Department of Physics and Centre for Quantum Computer Technology, University of Queensland, Brisbane 4072, Australia, 2Institute for Quantum Optics and Quantum Information, Austrian Academy of Sciences, Boltzmanng. 3, A-1090 Vienna, Austria, 3Institute for Quantum Computing and Department of Physics & Astronomy, University of Waterloo, N2L 3G1, Canada, 4Centre for Quantum Dynamics, Griffith University, Brisbane 4111, Australia, 5Centre for Quantum Photonics, H. H. Wills Physics Laboratory and Department of Electrical and Electronic Engineering, University of Bristol, Merchant Venturers Building, Woodland Road, Bristol BS8 1UB, UK, 6Physics Department, Macquarie University, Sydney 2109, Australia. *e-mail: [email protected]. 134 NATURE PHYSICS j VOL 5 j FEBRUARY 2009 j www.nature.com/naturephysics NATURE PHYSICS DOI: 10.1038/NPHYS1150 ARTICLES a C2 C3 T C 1 † † C T T Zπ 2 T C H T † T T † T H 1 1 0 T b C2 Qubit H H Xa Xb Xb Xa 0 1 C1 Qubit X 1 Qutrit operator T Qutrit Figure 2 j Simplifying higher-order Toffoli gates. Three-control-qubit H H Xa Xa 00 1 11 0 0 1 0 Toffoli . The Xa gate swaps information between the logical j0i and j2i Xa = 1 00 states of the target. The Xb gate flips information between the logical j1i j i Qubit operators and 3 state of the target. Thus, we require access to a four-level target 1 0 0 0 1 0 0 0 1 0 1 0 information carrier: two levels in the original rail and one in each of the T iπ/4 Zθ i θ 0 1 0 0 0 1 0 0 0 e 0 e dashed rails. The target undergoes a sign shift only for the input term 0 0 1 0 0 0 0 1 1 11 0 1 0 0 0 –1 1 H jC ;C ;C ;TiDj1;1;1;1i. This operation is equivalent to the Toffoli under the 0 0 0 2 1–1 X 1 0 3 2 1 action of only two one-qubit gates, as shown. See Fig. 1 for gate operations. Figure 1 j Simplifying the Toffoli gate. a, Most efficient known a decomposition into the universal gate set CNOT Carbitrary one-qubit gate, C1 when restricted to operating on qubits12. b, Our decomposition requiring 1 11 T † Z only three two-qubit gates . Here, the target is a three-level ‘qutrit’ with V U V Xa θ Xa Zθ logical states j0i, j1i and j2i. Initially and finally, all of the quantum 0 information is encoded in the j0i and j1i levels of each information carrier. b C2 The action of the X gates is to swap information between the logical j0i a C and j2i states of the target. The target undergoes a sign shift only for the 1 j iDj i 1 0 input term C2;C1;T 1;0;1 . This operation is equivalent to the Toffoli T V U V † Z Z under the action of only three one-qubit gates, as shown. The second gate Xa Xb θ Xb Xa θ 0 1 in the decomposition is a CZ and is equivalent to a CNOT under the action of two one-qubit Hadamard (H) gates. only 2n−1 standard two-qubit gates11; that is, with each extra Figure 3 j Simplifying controlled-unitary gates. a, One control qubit (we control qubit we need an extra level in the target carrier (see implement a simplified version, see Fig. 5): the control operation occurs if Fig. 2). Compare this with the previous best known scheme, which jC1iDj0i. b, Two control qubits: the control operation occurs if y requires 12n−11 two-qubit gates and an extra overhead of n−1 jC2;C1iDj1;1i. VZθ V is the spectral decomposition of U, up to a global extra ancilla qubits12. When restrained from using ancilla, this phase factor. See Fig. 1 for gate operations. scheme requires of the order of n2 two-qubit gates. In either case, we achieve a significant resource reduction, by harnessing only way in which the elemental gates are realized. Consequently, it higher levels of existing information carriers. For example, the has the potential for application in many architectures, yielding simplest known decomposition of the 5t requires 50 two-qubit gates the same resource savings. The only physical requirements are and four ancilla qubits, when restricted to operating on qubits12. access to multi-level systems and the ability to coherently swap Our technique requires only nine two-qubit gates and no ancillary information between these levels, that is, implement the generalized information carriers.