Bulk quantum computation with nuclear magnetic resonance: theory and experiment

Abstract

We show that quantum computation is possible with mixed states instead of pure states as inputs. This is performed by embedding within the mixed state a subspace that transforms like a pure state and that can be identified by labelling based on logical (spin), temporal, or spatial degrees of freedom. This permits quantum computation to be realized with bulk ensembles far from the ground state. Experimental results are presented for quantum gates and circuits implemented with liquid nuclear magnetic resonance techniques and verified by quantum state tomography.

Hadamard products of product operators and the design of gradient-diffusion experiments for simulating decoherence by NMR spectroscopy

“…This representation is obtained by expanding the matrices versus the "product operator" basis consisting of all possible N-fold Kronecker (or tensor) products of the usual Pauli matrices σ x , σ y , σ z and the 2 × 2 identity σ 1 . Together with the simpler and more general rules provided by the underlying geometric algebra structure [5,6], the product operator formalism has also proven invaluable in designing pulse sequences to implement a wide variety of logic gates for quantum information processing (QIP) by NMR [7][8][9][10] (see [11,12] for recent reviews).…”

Relaxation of pseudo pure states: the role of cross-correlations

“…Since the uniform background populations do not contribute to the NMR signal, such a state then mimics a pure state. Several methods of creating PPS have been developed like spatial averaging [5,6], logical labeling [7,8,9,10], temporal averaging [11], spatially averaged logical labeling technique (SALLT) [12]. However pseudo pure state, as well as pure states are not stationary and are destroyed with time as the spin system relaxes toward equilibrium.…”

Discrimination of unitary transformations in the Deutsch-Jozsa algorithm: Implications for thermal-equilibrium-ensemble implementations

“…In some proposed implementations of quantum computing, such as solution state nuclear magnetic resonance (NMR) [18][19][20][21][22][23][24][25][26][27], the initial state of the quantum system is mixed and therefore the problem cannot be solved with certainty by using the scheme of Fig. 1.…”

“…where ω i is the precession frequency of the i th qubit and J ij is the coupling between the i th and j th qubits (this is typical for solution state NMR [18,19]). The thermal equilibrium density operator isρ th = e −βĤ /Z where β = 1/kT and Z = Tr e −βĤ .…”