Quantum process tomography of a Mølmer-Sørensen interaction

Abstract

We report the quantum process tomography of a Mølmer-Sørensen entangling gate. The tomographic protocol relies on a single discriminatory transition, exploiting excess micromotion in the trap to realize all operations required to prepare all input states and analyze all output states. Using a master-slave diode lasers setup, we demonstrate a two-qubit entangling gate, with a fidelity of Bell state production of 0.985(10). We characterize its χ-process matrix, the simplest for an entanglement gate on a separable-states basis, and we observe that the dominant source of error is accurately modelled by a quantum depolarization channel. PACS numbers:The ability to realize and characterize high-fidelity two-qubit gates is central for quantum information science as, together with single-qubit rotations, they constitute the building blocks for quantum computation [1]. The detailed characterization of these gates is therefore crucial. Quantum Process Tomography (QPT) is an important method to fully characterize linear quantum processes. In particular, QPT of two-qubit entangling gates has been used to characterize CNOT gates in linear-optic [2], NMR [3], as well as trapped ions [4,5], or a square root i-SWAP gate with superconducting qubits [6]. In trapped-ions experiments, Mølmer-Sørensen (MS) entangling gates [7] have become increasingly popular, both for quantum computation purposes [8,9] and for inducing effective spin-spin couplings that allow to simulate complex quantum many-body hamiltonians from condensed matter physics [10]. One of its main advantages as compared with other gate protocols is its first-order insensitivity to the phonon occupation number (i.e. temperature of the ion-crystal), which allowed, inter alia, the highest entangled state production fidelity reached to date (0.993(1) [11]), entanglement between ions in thermal motion [12], as well as the creation of a maximally entangled state of a large (N = 14) number of qubits [13]. In this letter, we first implement a new and simple protocol for QPT with trapped ions, which only requires a single discriminatory transition. The scheme is based on inhomogeneous micromotion in the trap that enables addressing single qubits in the chain [14][15][16]. Subsequently, we realize the tomographic reconstruction of a Mølmer-Sørensen interaction which, despite its growing importance, has not been process-analyzed yet.A quantum process is defined as a completely positive map E in the space of density matrices. Given a complete set of operators {A i } (such that j A † j A j = I), the * These authors contributed equally to this work. † Present email and address: nn270@cam.ac.uk, Cavendish Laboratory, University of Cambridge, J.J. Thomson Ave., Cambridge CB3 0HE, United Kingdom output state for an arbitrary input state ρ can be written as (for details see for instance [3,17,18])(1)Here {χ ab } is the process matrix (with 4 n × 4 n elements for n qubits), which contains the full information on the process E and is measured by QPT. A convenient set of input states for t...

Universal gate-set for trapped-ion qubits using a narrow linewidth diode laser

“…The two operations discussed above: two-qubit collective rotations and individual addressing of only one qubit are sufficient in the case of a two-qubit register to perform any single qubit gate. As an example in [35] we have used this scheme to perform complete quantum process tomography for the MS two qubit gate. In principle the use of the spatially varying electric field and micromotion induced sidebands can be generalized to more than two ions by introducing the concept of dressed-state picture [34].…”

Multi-qubit gate with trapped ions for microwave and laser-based implementation

“…Simulating all the terms but equation (17) yields infidelity of = × − IF 3 10 4 ( figure 5), getting closer to the fault tolerance threshold. For these suggested parameters, the contributions of the coupling terms, equations (23) and (24), correspond to infidelity of × − 1.2 10 5 and × − 5 10 5 , respectively.…”

“…Many theoretical proposals for quantum entangling gates have been considered for trapped ions [2][3][4][5][6][7][8][9][10][11], one of the most promising candidates for QIP [12][13][14]. These proposals have triggered impressive experimental realizations [15][16][17][18][19][20][21][22][23][24][25]. Although these experiments within the laser based designs have achieved high fidelities [16,[22][23][24][25], the achieved fidelities within the microwave based designs have been limited.Considerable theoretical efforts have been made to counter the fidelity-damaging effects.…”

“…These proposals have triggered impressive experimental realizations [15][16][17][18][19][20][21][22][23][24][25]. Although these experiments within the laser based designs have achieved high fidelities [16,[22][23][24][25], the achieved fidelities within the microwave based designs have been limited.Considerable theoretical efforts have been made to counter the fidelity-damaging effects. Techniques, such as dynamical decoupling using echo-pulse sequences [26,27], and its continuous version using a continuously applied driving field [28-33], have been proposed and realized experimentally [20,[34][35][36].…”

Enhancing the fidelity of two-qubit gates by measurements

“…Entangling gates with trapped ions-Entangling gates with trapped ions are realized using the Mølmer-Sørensen (MS) scheme [13][14][15][16][17][18][19][20][21], which is constructed out of the following interaction: vibrational phonon, Ω is the sideband Rabi frequency, and ε = ω d − ν is the detuning of the driving field from the secular frequency. During the MS gate operation, the two spins are entangled to the vibrational phonon, which generates the entanglement between the two spins.…”