Relations between the single-pass and multi-pass qubit gate errors

In quantum computation the target fidelity of the qubit gates is very high, with the admissible error being in the range from $10^{-3}$ to $10^{-4}$ and even less, depending on the protocol. The direct experimental determination of such an extremely small error is very challenging by standard quantum-process tomography. Instead, the method of randomized benchmarking, which uses a random sequence of Clifford gates, has become a standard tool for determination of the average gate error as the decay constant in the exponentially decaying fidelity. In this paper, the task for determining a tiny error is addressed by sequentially repeating the \emph{same} gate multiple times, which leads to the coherent amplification of the error, until it reaches large enough values to be measured reliably. If the transition probability is $p=1-\epsilon$ with $\epsilon \ll 1$ in the single process, then classical intuition dictates that the probability after $N$ passes should be $P_N \approx 1 - N \epsilon$. However, this classical expectation is misleading because it neglects interference effects. This paper presents a rigorous theoretical analysis based on the SU(2) symmetry of the qubit propagator, resulting in explicit analytic relations that link the $N$-pass propagator to the single-pass one in terms of Chebyshev polynomials. In particular, the relations suggest that in some special cases the $N$-pass transition probability degrades as $P_N = 1-N^2\epsilon$, i.e. dramatically faster than the classical probability estimate. In the general case, however, the relation between the single-pass and $N$-pass propagators is much more involved. Recipes are proposed for unambiguous determination of the gate errors in the general case, and for both Clifford and non-Clifford gates.Comment: 9 pages, 5 figure

Relations between the single-pass and double-pass transition probabilities in quantum systems with two and three states

In the experimental determination of the population transfer efficiency between discrete states of a coherently driven quantum system it is often inconvenient to measure the population of the target state. Instead, after the interaction that transfers the population from the initial state to the target state, a second interaction is applied which brings the system back to the initial state, the population of which is easy to measure and normalize. If the transition probability is $p$ in the forward process, then classical intuition suggests that the probability to return to the initial state after the backward process should be $p^2$. However, this classical expectation is generally misleading because it neglects interference effects. This paper presents a rigorous theoretical analysis based on the SU(2) and SU(3) symmetries of the propagators describing the evolution of quantum systems with two and three states, resulting in explicit analytic formulas that link the two-step probabilities to the single-step ones. Explicit examples are given with the popular techniques of rapid adiabatic passage and stimulated Raman adiabatic passage. The present results suggest that quantum-mechanical probabilities degrade faster in repeated processes than classical probabilities. Therefore, the actual single-pass efficiencies in various experiments, calculated from double-pass probabilities, might have been greater than the reported values.Comment: 8 pages, 5 figure

Robust high-fidelity coherent control of two-state systems by detuning pulses

Coherent control of two-state systems is traditionally achieved by resonant pulses of specific Rabi frequency and duration, by adiabatic techniques using level crossings or delayed pulses, or by sequences of pulses with precise relative phases (composite pulses). Here we develop a method for high-fidelity coherent control which uses a sequence of detuning pulses. By using the detuning pulse areas as control parameters, and driving on an analogy with composite pulses, we report a great variety of detuning pulse sequences for broadband and narrowband transition probability profiles.Comment: 8 pages, 9 figure

Achromatic multiple beam splitting by adiabatic passage in optical waveguides

A novel variable achromatic optical beam splitter with one input and $N$ output waveguide channels is introduced. The physical mechanism of this multiple beam splitter is adiabatic passage of light between neighboring optical waveguides in a fashion reminiscent of the technique of stimulated Raman adiabatic passage in quantum physics. The input and output waveguides are coupled via a mediator waveguide and the ratios of the light intensities in the output channels are controlled by the couplings of the respective waveguides to the mediator waveguide. Due to its adiabatic nature the beam splitting efficiency is robust to variations in the experimental parameters

Spin splitting of relativistic particles in 3D

The behavior of relativistic particles in an electric and/or magnetic field is considered in the general case when the direction of propagation may differ from the direction of the field. A special attention is paid to the spin splitting and the ensuing Larmor precession frequency of both neutral and charged particles. For both neutral and charged particles, the Larmor frequency shows a longitudinal motional red shift. For a neutral particle, there is a dynamical upper bound, which depends on both the mass and the transverse momentum of the particle; moreover, the transverse motion leads to a blue shift of the Larmor frequency. For a charged particle, the longitudinal motional decrease of the spin splitting is determined by the formation of Landau levels and it has no upper limit. Unlike the nonrelativistic limit, the relativistic spin splitting depends on the Landau levels and decreases for higher Landau levels, thereby signalling the presence of a Landau ladder red shift effect

Relativistic effects for spin splitting of neutral particles: Upper bound and motional narrowing

We explore the properties of spin splitting for neutral particles possessing electric and magnetic dipole moments propagating in an electromagnetic field. Two notable features of the spin splitting and the associated Larmor precession are found, which are consequences of special relativity. First, we report the existence of an upper limit of spin splitting equal to twice the rest energy of the particle, and a corresponding upper limit for the Larmor precession frequency. Second, we predict the noninvariance of the spin splitting and the corresponding Larmor frequency with respect to Lorentz boosts, which bears resemblance to the classical Doppler effect

High-fidelity multistate STIRAP assisted by shortcut fields

Multistate stimulated Raman adiabatic passage (STIRAP) is a process which allows for adiabatic population transfer between the two ends of a chainwise-connected quantum system. The process requires large temporal areas of the driving pulsed fields (pump and Stokes) in order to suppress the nonadiabatic couplings and thereby to make adiabatic evolution possible. To this end, in the present paper a variation of multistate STIRAP, which accelerates and improves the population transfer, is presented. In addition to the usual pump and Stokes fields it uses shortcut fields applied between the states, which form the dark state of the system. The shortcuts cancel the couplings between the dark state and the other adiabatic states thereby resulting (in the ideal case) in a unit transition probability between the two end states of the chain. Specific examples of five-state systems formed of the magnetic sublevels of the transitions between two degenerate levels with angular momenta $J_g=2$ and $J_e=1$ or $J_e=2$ are considered in detail, for which the shortcut fields are derived analytically. The proposed method is simpler than the usual "shortcuts to adiabaticity" recipe, which prescribes shortcut fields between all states of the system, while the present proposal uses shortcut fields between the sublevels forming the dark state only. The results are of potential interest in applications where high-fidelity quantum control is essential, e.g. quantum information, atom optics, formation of ultracold molecules, cavity QED, etc.Comment: 11 pages, 10 figure

Composite stimulated Raman adiabatic passage

We introduce a high-fidelity technique for coherent control of three-state quantum systems, which combines two popular control tools --- stimulated Raman adiabatic passage (STIRAP) and composite pulses. By using composite sequences of pairs of partly delayed pulses with appropriate phases the nonadiabatic transitions, which prevent STIRAP from reaching unit fidelity, can be canceled to an arbitrary order by destructive interference, and therefore the technique can be made arbitrarily accurate. The composite phases are given by simple analytic formulas, and they are universal for they do not depend on the specific pulse shapes, the pulse delay and the pulse areas.Comment: 5 pages, 5 figure

Composite two-qubit gates

We design composite controlled-phase gates, which compensate errors in the phase of a single gate. The errors can be of various nature, such as relative, absolute or both. We present composite sequences which are robust to relative errors up to the 6th order, with the number of the constituent gates growing just linearly with the desired accuracy, and we describe a method to achieve even higher accuracy. We show that the absolute error can be canceled entirely with only two gates. We describe an ion-trap implementation of our composite gates, in which simultaneous cancellation of the error in both the pulse area and the detuning is achieved

High-fidelity error-resilient composite phase gates

We present a method to construct high-fidelity quantum phase gates, which are insensitive to errors in various experimental parameters. The phase gates consist of a pair of two sequential broadband composite pulses, with a phase difference $\pi+\alpha/2$ between them, where $\alpha$ is the desired gate phase. By using composite pulses which compensate systematic errors in the pulse area, the frequency detuning, or both the area and the detuning, we thereby construct composite phase gates which compensate errors in the same parameters. Particularly interesting are phase gates which use the recently discovered universal composite pulses, which compensate systematic errors in any parameter of the driving field, which keep the evolution Hermitian (e.g., pulse amplitude and duration, pulse shape, frequency detuning, Stark shifts, residual frequency chirps, etc.Comment: 5 pages, 4 figure