Universal fault-tolerant quantum computers will require error-free execution of long sequences of quantum gate operations, which is expected to involve millions of physical qubits. Before the full power of such machines will be available, near-term quantum devices will provide several hundred qubits and limited error correction. Still, there is a realistic prospect to run useful algorithms within the limited circuit depth of such devices. Particularly promising are optimization algorithms that follow a hybrid approach: the aim is to steer a highly entangled state on a quantum system to a target state that minimizes a cost function via variation of some gate parameters. This variational approach can be used both for classical optimization problems as well as for problems in quantum chemistry. The challenge is to converge to the target state given the limited coherence time and connectivity of the qubits. In this context, the quantum volume as a metric to compare the power of near-term quantum devices is discussed.With focus on chemistry applications, a general description of variational algorithms is provided and the mapping from fermions to qubits is explained. Coupledcluster and heuristic trial wave-functions are considered for efficiently finding molecular ground states. Furthermore, simple error-mitigation schemes are introduced that could improve the accuracy of determining ground-state energies. Advancing these techniques may lead to near-term demonstrations of useful quantum computation with systems containing several hundred qubits.PACS numbers: quantum computation, quantum chemistry, quantum algorithms

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Spin-orbit coupling is a manifestation of special relativity. In the reference frame of a moving electron, electric fields transform into magnetic fields, which interact with the electron spin and lift the degeneracy of spin-up and spin-down states. In solid-state systems, the resulting spin-orbit fields are referred to as Dresselhaus or Rashba fields, depending on whether the electric fields originate from bulk or structure inversion asymmetry, respectively. Yet, it remains a challenge to determine the absolute value of both contributions in a single sample. Here we show that both fields can be measured by optically monitoring the angular dependence of the electrons' spin precession on their direction of movement with respect to the crystal lattice. Furthermore, we demonstrate spin resonance induced by the spin-orbit fields. We apply our method to GaAs/InGaAs quantum-well electrons, but it can be used universally to characterise spin-orbit interactions in semiconductors, facilitating the design of spintronic devices.Symmetry-breaking electric fields in semiconductors induce a spin splitting, because electric fields appear to a moving electron as magnetic fields, which interact with the electron spin and couple it with the electron momentum, or wave vector, k. In zinc-blende-type crystals, such as GaAs, the electric fields resulting from the lack of an inversion centre lead to bulk inversion asymmetry (BIA) and to the Dresselhaus term in the Hamiltonian [1]. In the conduction band, its coupling is linear or cubic in k with proportionality constants β and γ, respectively. In heterostructures, additional electric fields are introduced owing to structure inversion asymmetry (SIA), giving rise to the Rashba term [2], which for conduction-band electrons is linear in k with coupling constant α. Both contributions have been extensively studied [3], since a potential use of electron spins in future devices (e.g. a spin transistor [4]) requires precise control of the spin's environment and of the Dresselhaus and Rashba fields [5]. Spin-orbit fields also contribute to spin decoherence [6].In two-dimensional systems, such as quantum wells (QWs), usually α β and γ ≈ 0 [7,8,9,10]. Therefore, measurements of the spin-orbit coupling initially focused on the Rashba term in QWs and concentrated on the study of beatings in Shubnikov-de-Haas oscillations [8,10,11,12,13], whose interpretation, however, is debated [14,15]. More recent experiments include the investigation of antilocalization in magnetotransport [16] or the analysis of photocurrents [17]. In the latter experiment, the ratio α/β could be determined. A gateinduced transition from weak localization to antilocalization allowed the discrimination between Rashba, as well as linear and cubic Dresselhaus contributions to the spinorbit field [18]. Tuning of the Rashba coupling has been achieved by introducing additional electric fields from gates [9,19] or by changing the electron density [20,21].The influence of effective spin-orbit magnetic fields on optical measurements in ...

The spin-orbit interaction (SOI) in zincblende semiconductor quantum wells can be set to a symmetry point, in which spin decay is strongly suppressed for a helical spin mode. Signatures of such a persistent spin helix (PSH) have been probed using the transient spin grating technique, but it has not yet been possible to observe the formation and the helical nature of a PSH. Here we directly map the diffusive evolution of a local spin excitation into a helical spin mode by a timeand spatially resolved magneto-optical Kerr rotation technique. Depending on its in-plane direction, an external magnetic field interacts differently with the spin mode and either highlights its helical nature or destroys the SU(2) symmetry of the SOI and thus decreases the spin lifetime. All relevant SOI parameters are experimentally determined and confirmed with a numerical simulation of spin diffusion in the presence of SOI.Conduction-band electrons in semiconductors experience SOI from intrinsic [1] and extrinsic sources, leading to spin dephasing, current-induced spin polarization and spin Hall effects [2]. These physical mechanisms are of great fundamental and technological interest, recently also in the context of topolocial insulators [3] and Majorana fermions [4,5]. Intrinsic SOI arises from an inversion asymmetry of the bulk crystal (Dresselhaus term) and of the grown layer structure (Rashba term). In a quantum well (QW), these two components can be tailored by means of the confinement potential [6], and the Rashba SOI can be externally tuned by using gate electrodes [7,8]. In general, SOI leads to precession of electron spins. In the diffusive limit, in which the scattering length is much smaller than the spin-orbit (SO) length λ SO , a random walk of the spins on the Bloch sphere will dephase a non-equilibrium spin polarization [9].Of special interest is the situation in a two-dimensional electron gas (2DEG) with balanced Rashba and Dresselhaus contributions [6,[10][11][12][13]. There, the SOI attains SU(2) symmetry and the spin polarization of a helical mode is preserved. The reason for this conservation of the spin polarization is a unidirectional effective SO magnetic field B SO , which depends linearly on the component of the electron momentum along a specific in-plane direction. This causes the precession angle of a moving electron to vary linearly with the distance traveled along that direction, irrespective of whether the electron path is ballistic or diffusive [10,11]. In such a situation, a local spin excitation is predicted to evolve into a helical spin mode termed PSH [ Fig. 1(a)]. Transient spin grating measurements [6] showed that a spin excitation with a spatially modulated out-of-plane spin component decays with two characteristic lifetimes that correspond to two superposed spin modes of opposite helicity. * Electronic address: gsa@zurich.ibm.comHere we directly measure the diffusive evolution of a local spin excitation into a PSH by time-resolved Kerr rotation microscopy [ Fig. 1(b)]. We employ a pumpprobe appro...

The processing of quantum information based on the electron spin degree of freedom requires fast and coherent manipulation of local spins. One approach is to provide spatially selective tuning of the spin splitting--which depends on the g-factor--by using magnetic fields, but this requires their precise control at reduced length scales. Alternative proposals employ electrical gating and spin engineering in semiconductor heterostructures involving materials with different g-factors. Here we show that spin coherence can be controlled in a specially designed AlxGa1-xAs quantum well in which the Al concentration x is gradually varied across the structure. Application of an electric field leads to a displacement of the electron wavefunction within the quantum well, and because the electron g-factor varies strongly with x, the spin splitting is therefore also changed. Using time-resolved optical techniques, we demonstrate gate-voltage-mediated control of coherent spin precession over a 13-GHz frequency range in a fixed magnetic field of 6 T, including complete suppression of precession, reversal of the sign of g, and operation up to room temperature.

A key requirement to perform simulations of large quantum systems on near-term quantum hardware is the design of quantum algorithms with short circuit depth that finish within the available coherence time. A way to stay within the limits of coherence is to reduce the number of gates by implementing a gate set that matches the requirements of the specific algorithm of interest directly in hardware. Here, we show that exchange-type gates are a promising choice for simulating molecular eigenstates on near-term quantum devices since these gates preserve the number of excitations in the system. Complementing the theoretical work by Barkoutsos et al. [PRA 98, 022322 (2018)], we report on the experimental implementation of a variational algorithm on a superconducting qubit platform to compute the eigenstate energies of molecular hydrogen. We utilize a parametrically driven tunable coupler to realize exchange-type gates that are configurable in amplitude and phase on two fixed-frequency superconducting qubits. With gate fidelities around 95% we are able to compute the eigenstates within an accuracy of 50 mHartree on average, a limit set by the coherence time of the tunable coupler.The simulation of the electronic structure of molecular and condensed matter systems is a challenging computational task as the cost of resources increases exponentially with the number of electrons when accurate solutions are required. With the tremendous improvements in our ability to control complex quantum systems this bottleneck may be overcome by the use of quantum computing hardware [1]. In theory, various algorithms for quantum simulation have been designed to that end, including quantum phase estimation [2] or adiabatic algorithms [3]. With these algorithms the challenges for practical applications lie in the efficient mapping of the electronic Hamiltonian onto the quantum computer and in the required number of quantum gates that remains prohibitive on current and near-term quantum hardware [4] without quantum error correction schemes [5]. On the other hand, variational quantum eigensolver (VQE) methods [6, 7] can produce accurate results with a small number of gates [8] using for instance algorithms with low circuit depth [9] and do not require a direct mapping of the electronic Hamiltonian onto the hardware. Moreover, such algorithms are inherently robust against certain errors [8, 10, 11] and are therefore considered as ideal candidates for first practical implementations on non error-corrected, near-term quantum hardware.Recently, the molecular ground state energy of hydrogen and helium have been computed via VQE in proof of concept experiments using NMR quantum simulators [12][13][14], photonic architectures [6] or nitrogenvacancy centers in diamond [15]. Although very accurate energy estimates are obtained, quantum simulation of larger systems remains an intractable problem on these platforms because of the difficulties arising in scaling them up to more than a few qubits. For this reason trapped ions [16][17][18][19] and supe...

Conduction electrons are used to optically polarize, detect and manipulate nuclear spin in a (110) GaAs quantum well. Using optical Larmor magnetometry, we find that nuclear spin can be polarized along or against the applied magnetic field, depending on field polarity and tilting of the sample with respect to the optical pump beam. Periodic optical excitation of the quantum-confined electron spin reveals a complete spectrum of optically-induced and quadrupolar-split nuclear resonances, as well as evidence for ∆m = 2 transitions.

Time-resolved optical measurements of electron-spin dynamics in a (110) GaAs quantum well are used to study the consequences of a strongly anisotropic electron gtensor, and the origin of previously discovered all-optical nuclear magnetic resonance.All components of the g-tensor are measured, and a strong anisotropy even along the inplane directions is found. The amplitudes of the spin signal allow the study of the spatial directions of the injected spin and its precession axis. Surprisingly efficient dynamic nuclear polarization in a geometry where the electron spins are injected almost transverse to the applied magnetic field is attributed to an enhanced non-precessing electron spin component. The small absolute value of the electron g-factor combined with efficient nuclear spin polarization leads to large nuclear fields that dominate electron spin precession at low temperatures. These effects allow for sensitive detection of all-optical nuclear magnetic resonance induced by periodically excited quantum-well electrons. The mechanism of previously observed ∆m = 2 transitions is investigated and found to be attributable to electric quadrupole coupling, whereas ∆m = 1 transitions show signatures of both quadrupole and electron-spin induced magnetic dipole coupling.