Quantum Process Tomography of a Mlmer-Srensen Interaction
We report the quantum process tomography of a Mlmer-Srensen 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 -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.
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 Barenco et al. (1995). 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 Kiesel et al. (2005), NMR Childs et al. (2001), as well as trapped ions Riebe et al. (2006); Home et al. (2009), or a square root i-SWAP gate with superconducting qubits Bialczak et al. (2010). In trapped-ions experiments, Mlmer-Srensen (MS) entangling gates Sørensen and Mølmer (1999) have become increasingly popular, both for quantum computation purposes Sackett et al. (2000); Leibfried et al. (2003) and for inducing effective spin-spin couplings that allow to simulate complex quantum many-body hamiltonians from condensed matter physics Kim et al. (2009). 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 ( Benhelm et al. (2008)), entanglement between ions in thermal motion Kirchmair et al. (2009), as well as the creation of a maximally entangled state of a large () number of qubits Monz et al. (2011). 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 Turchette et al. (1998); Warring et al. (2012); Navon et al. (2012). Subsequently, we realize the tomographic reconstruction of a Mlmer-Srensen interaction which, despite its growing importance, has not been process-analyzed yet.
A quantum process is defined as a completely positive map in the space of density matrices. Given a complete set of operators (such that ), the output state for an arbitrary input state can be written as (for details see for instance Poyatos et al. (1997); Nielsen and Chuang (2010); Childs et al. (2001))
Here is the process matrix (with elements for qubits), which contains the full information on the process and is measured by QPT. A convenient set of input states for the tomography is the product states , where , which are the one-qubit eigenstates of the Pauli matrices ,,, with eigenvalues . Note that, with this choice, entangled states are not used as input states. The measurement basis is conveniently chosen to be where , and . However, in the experiment, the detection scheme relies on the statistics of fluorescence photons, which corresponds to the measurement of the expectation value . In order to measure the expectation value of , we perform additional rotations on the two qubits. In general these rotations require single-qubit addressing capability. For our purpose, a single discriminatory transition is sufficient for all the required operations.
In our setup, we use Sr ions confined in a linear Paul trap Akerman et al. (2011). We work with optical qubits that are encoded in the ground state level and in the meta-stable level which has a lifetime of ms Letchumanan et al. (2005). Coherent manipulation of the qubit state is performed with a narrow linewidth laser at 674 nm which drives an electric-quadrupole transition Dio (). The other Zeeman level of the ground state , separated by MHz from the level due to a constant magnetic field, is used as auxiliary level in the state detection scheme. Measuring the qubit state is accomplished by counting fluorescence photons on the dipole transition with a photomultiplier tube (PMT). We inferred the number of ions in the (bright) state, for each realization, by the number of detected photons. The probabilities, , , , of finding zero, one and two ions in the (bright) state were estimated by the fraction of realizations with the corresponding number of ions inferred in that state. The discriminatory transition is provided through a micromotion sideband. In an ideal linear Paul trap, the symmetry axis of the trap is also the axis where the RF vanishes, and no excess micromotion is present. However, due to the finite size of the trap, the boundary conditions set by the endcaps leads to an rf leak along this axis, and the region of rf-null is reduced to a point at the center. If the two-ion chain is axially aligned so that one ion sits on the rf-null, the other ion is the only one to possess micromotion sidebands, on which selective quantum control can be performed Leibfried (1999). Axial displacement of the ion-crystal is realized by applying a differential voltage on the two endcaps. In the limit of small amplitudes, the Rabi frequency of the sideband is where is the carrier Rabi frequency, is the micromotion Lamb-Dicke parameter and is the micromotion amplitude along the laser wavevector . To maintain coherence throughout the experimental sequence, the trap rf and the signals feeding the acousto-optic modulator (AOM) that drive both the micromotion sideband and the carrier transition, are all phase-locked to the same time base.
Our protocol for implementing two qubit QPT is illustrated in Fig.1a.
In order to measure in all the necessary bases, it is enough to possess single-qubit rotation capability on one qubit only. We look for an operation such that . Indeed, can be decomposed in the form: , where () is a global (local) rotation around a direction lying in the -plane of the optical qubit. More precisely, these operators can be written as , a global rotation around the -axis, and a local rotation around the -axis of only one ion (where ). For example , , , and so on. Similarly, the state preparation of all product states mentioned above can be realized using the same set of operations after initializing the ions to by optical pumping. Lastly, the value of is extracted from fluorescence histograms. In addition, some of the necessary measurements for QPT are of the form or , and thus require the measurement of the state of each ion separately. To perform these measurements we utilize the auxiliary level to which we transfer one of the ions into a definitely bright state (). This is accomplished by first transferring the state population into in both ions with an rf -pulse. Then another -pulse on the micromotion sideband transfers the state population of that ion into . The state of the ion at the null is then determined by (). Similarly, the state of the ion with micromotion is determined by applying an additional global carrier -pulse to both ions.
Using all the above, we first validate our QPT toolbox by characterizing the identity process, which amounts to concatenating the state preparation and analysis protocols without any intermediate operation. While the quantum state tomography of a two-qubit system requires for each input state 15 independent real-valued parameters (since is hermitian and ), a full QPT requires a total measurements for a system of qubits. From the measurement of the 16 output density matrices, we reconstruct the -matrix Nielsen and Chuang (2010). Due to noise and systematic errors in the measurements, the algebraically calculated -matrix is not physical i.e does not represent a completely positive map. We obtained a physical -matrix by means of maximum likelihood process estimation Hradil (1997). Fig.1b. shows the absolute value of the resulting process matrix. Here, the values of the imaginary part are small (). The definition of proper (and simple) distance measures for quantum operations is a subtle problem Gilchrist (2005). For simplicity, we will quantify the proximity between a tomographically reconstructed process and a target process by the mean fidelity: , where is the fidelity between the density matrices and , and the overline indicates average over all possible pure input states . Note that if the target process is unitary, then in the case of a pure input state, the corresponding output state, , is also pure, and therefore the fidelity takes the simpler form . Interestingly, this fidelity (contrary to the trace fidelity Nielsen and Chuang (2010)) is unity if and only if regardless of whether the processes are unitary or not. For the identity process of Fig.1, we find a mean fidelity of . While a single Rabi flop on the micromotion sideband was performed with a fidelity of , slow drifts of stray fields lead to small displacements of the ion from the rf-null. Furthermore small laser detuning errors reduced the single-qubit rotation fidelity.
Next we apply our tomography protocol to analyze a Mlmer-Srensen interaction. The gate is performed on the transition via two sidebands, which are generated by applying two rf signals into an acousto-optic modulator (AOM) switch Roos (2008), with frequencies of , where is the carrier transition frequency, is a motional sideband used for the intermediate spin-motion entanglement and is the gate detuning. We use the stretch axial mode at a frequency of MHz for entanglement as it is less sensitive to heating than the center of mass mode. The gate detuning is optimally set according to the Rabi frequency, , as , where is the motional Lamb-Dicke parameter, and the gate time is . After ground-state cooling of the stretch mode, the two ions are initialized by optical pumping to . The gate generates the maximally entangled state . In Fig.2a, we display the evolution of (blue points), (black points), and (red points), obtained for a gate detuning of kHz. At a pulse time of 130 s (shown by a vertical dashed green line), the two ions are maximally entangled. Together with parity analysis shown in Fig.2b, we measure a fidelity of the Bell-state production of .
After calibrating the gate time, we experimentally reconstructed the -process matrices of a single, three and five consecutive MS gates (the latter is shown in the upper panel of Fig.3). Each experiment is repeated 400 times, totaling measurements for a full process tomography. The target process matrices can be readily deduced from the evolution operator Roos (2008). Starting from the hamiltonian describing the ions interacting with the bichromatic field, one can show that at the gate time, the evolution operator reduces to . The motional part of the evolution operator is the identity, and only at the gate time (and multiples of it) the internal and motional parts factorize, leading to no loss of coherence for the internal-part due to the tracing of the motional degrees of freedom. The -matrix is calculated by expanding the exponent of : . Plugging this expression in Eq.(1), we readily find,
Interestingly, this operation matrix, with four non-zero elements, is considerably simpler in the -operators basis than the previously process-analyzed CNOT gates or iSWAP gates (with sixteen non-null elements each Childs et al. (2001); Bialczak et al. (2010)). The dominance of the process matrix elements in Eq.(2) is in good agreement with our results, shown in Fig.3.
The mean fidelity of with respect to are shown by blue points in Fig.3a. As seen, due to gate imperfections the fidelity decreases with growing number of applied gates at a rate of per gate. The direct interpretation of imperfections from the process matrix is notoriously difficult because in the operators basis each noise process involves multiple elements with various weights. Instead, it is common to compare the measured process to different noise models Childs et al. (2001); Bialczak et al. (2010); Kofman (2009). We found that the dominant error of our gate is consistent with a quantum depolarization channel for the two ions Nielsen and Chuang (2010), whose map is , where , and , is the depolarization rate. The single free parameter of the model can be first determined from the purity of our measured processes, which is affected only by non-unitary operations. We recall that the purity of state is . The rate of depolarization is determined by matching the slope (with respect to the number of gates) of (red points in Fig.3b) with the experimental purities (blue points in Fig.3b). The slope matches our data for a rate of per gate time. The identity process is excluded from these fits, since errors from tomography and the gates have a different origin, they are largely independent.
We can verify the appropriateness of this description by calculating the fidelity of with respect to an MS interaction that has suffered partial depolarization (red points of Fig.3b), where (). Remarkably, we find that for the rate previously determined from the averaged purity, the mean fidelity to the partially depolarized states is almost constant as a function of the number of gates applied. This shows that the depolarization channel accounts well for the imperfections introduced by successive applications of gates, and we conclude that the remaining error is due to the tomography itself. Morever, we can use the depolarization channel model to predict the expected imperfections on the population time dynamics previously measured in Fig.2. On one hand, these contain much more limited information than the full process matrices, namely only the diagonal elements of the output state starting from , but on the other hand, they are free from tomographic errors. While the solution of the perfect MS propagator (dashed lines in Fig.2) does not describe our data well for the longest times, the agreement is excellent when the depolarization is taken into account (solid lines), especially since there is no adjustable parameter. In particular, at we expect a Bell state production fidelity of , in very good agreement with the experimental determination. While the physical origin of the depolarization is unknown to us, the measured depolarization rate is in rough agreement with off-resonance incoherent transfer rate we observe on a single trapped ion and that is generated by fast (1 MHz) phase noise of our laser. A more thorough study of the cause for depolarization is under way.
In conclusion we implemented a simple method for QPT of two-qubit processes based on a single discriminatory transition and with no direct spatially-selective imaging. The protocol was used to tomographically reconstruct a high fidelity Mlmer-Srensen interaction. The Mlmer-Srensen interaction is currently the main method for generating entangling gates with trapped ion qubits and for synthesizing coupling between trapped ion spins for quantum simulation and this work provides the first full characterization of its process matrix.
This research was supported by the Israeli Science Foundation, the Minerva Foundation, the German-Israeli Foundation for Scientific Research, the Crown Photonics Center, the Wolfson Family Charitable Trust, Yeda-Sela Center for Basic Research, David Dickstein of France and M. Kushner Schnur, Mexico.
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