A major challenge in quantum computing is to solve general problems with limited physical hardware. Here, we implement digitized adiabatic quantum computing, combining the generality of the adiabatic algorithm with the universality of the digital approach, using a superconducting circuit with nine qubits. We probe the adiabatic evolutions, and quantify the success of the algorithm for random spin problems. We find that the system can approximate the solutions to both frustrated Ising problems and problems with more complex interactions, with a performance that is comparable. The presented approach is compatible with small-scale systems as well as future error-corrected quantum computers.Quantum mechanics can help solve complex problems in physics [1], chemistry [2], and machine learning [3], provided they can be programmed in a physical device. In adiabatic quantum computing [4][5][6], the system is slowly evolved from the ground state of a simple initial Hamiltonian to a final Hamiltonian that encodes a computational problem. The appeal of this analog method lies in its combination of simplicity and generality; in principle, any problem can be encoded. In practice, applications are restricted by limited connectivity, available interactions, and noise. A complementary approach is digital quantum computing, where logic gates combine to form quantum circuit algorithms [7]. The digital approach allows for constructing arbitrary interactions and is compatible with error correction [8, 9], but requires devising tailor-made algorithms. Here, we combine the advantages of both approaches by implementing digitized adiabatic quantum computing in a superconducting system. We tomographically probe the system during the digitized evolution, explore the scaling of errors with system size, and measure the influence of local fields. We conclude by having the full system find the solution to random Ising problems with frustration, and problems with more complex interactions. This digital quantum simulation [10][11][12][13] consists of up to nine qubits and up to 10 3 quantum logic gates. This demonstration of digitized quantum adiabatic computing in the solid state opens a path to solving complex problems, and we hope it will motivate further research into the efficient synthesis of adiabatic algorithms, on small-scale systems with noise as well as future large-scale quantum computers with error correction.A key challenge in adiabatic quantum computing is to construct a device that is capable of encoding problem Hamiltonians that are non-stoquastic [14]. Such Hamiltonians would allow for universal adiabatic quantum computing [15, 16] as well as improving the performance for difficult instances * Present address: IBM T. J. Watson Research Center, Yorktown Heights, NY 10598, USA of classical optimization problems [17]. Additionally, simulating interacting fermions for physics and chemistry requires non-stoquastic Hamiltonians [1, 18]. In general, nonstoquastic Hamiltonians are more difficult to study classically, as Monte Carlo ...

Quantum tunnelling is a phenomenon in which a quantum state traverses energy barriers higher than the energy of the state itself. Quantum tunnelling has been hypothesized as an advantageous physical resource for optimization in quantum annealing. However, computational multiqubit tunnelling has not yet been observed, and a theory of co-tunnelling under high- and low-frequency noises is lacking. Here we show that 8-qubit tunnelling plays a computational role in a currently available programmable quantum annealer. We devise a probe for tunnelling, a computational primitive where classical paths are trapped in a false minimum. In support of the design of quantum annealers we develop a nonperturbative theory of open quantum dynamics under realistic noise characteristics. This theory accurately predicts the rate of many-body dissipative quantum tunnelling subject to the polaron effect. Furthermore, we experimentally demonstrate that quantum tunnelling outperforms thermal hopping along classical paths for problems with up to 200 qubits containing the computational primitive.

A new post-Markovian quantum master equation is derived, that includes bath memory effects via a phenomenologically introduced memory kernel k(t). The derivation uses as a formal tool a probabilistic single-shot bath-measurement process performed during the coupled system-bath evolution. The resulting analytically solvable master equation interpolates between the exact Nakajima-Zwanzig equation and the Markovian Lindblad equation. A necessary and sufficient condition for complete positivity in terms of properties of k(t) is presented, in addition to a prescription for the experimental determination of k(t). The formalism is illustrated with examples. An open quantum system is one that is coupled to an external environment [1,2]. Such systems are of fundamental interest, as the notion of a closed system is always an idealization and approximation. Open quantum systems tend to decohere, and for this reason have recently received intense consideration in quantum information science, where decoherence is viewed as fundamental obstacle to the construction of quantum information processors [3]. It is possible to write down an exact dynamical equation for an open system, but the resultan integro-differential equation [4] -is mostly of formal interest, as such an exact equation can almost never be solved analytically or even numerically. In contrast, when one makes the Markovian approximation, i.e., when one neglects all bath memory effects, the resulting Lindblad master equation [2,5] is formally solvable and amenable to numerical treatment. Moreover, the desirable property of complete positivity [6] is maintained (see, however, [7] for a debate on the importance of this property). A coveted goal of the theory of open quantum systems [1, 2] is a "post-Markovian" master equation that (i) generalizes the Markovian Lindblad equation so as to include bath memory effects, at the same time (ii) remains both analytically and numerically tractable, and (iii) retains complete positivity. A variety of post-Markovian master equations have been proposed and analyzed, e.g., [1,8,9,10,11,12,13,14,15,16]. However, one of the desirable properties (i)-(iii) above is typically lost: e.g., in the case of time-convolutionless master equations (e.g., [10]) one may lose complete positivity, while in the case of nonlocal stochastic Schrodinger equations (e.g., [13]) one loses analytical solvability. In this work we propose a new post-Markovian master equation that satisfies all of the desirable properties (i)-(iii) above. The key idea we introduce is an interpolation between the generalized measurement interpretation of the exact Kraus operator sum map [6], and the continuous measurement interpretation of Markovian-limit dynamics [16,18].Review of quantum measurements approach to open system dynamics.-Consider a quantum system S coupled to a bath B (with respective Hilbert spaces H S , H B ), evolving unitarily under the total system-bath Hamiltonian H SB . The exact system dynamics is given by tracing over the bath degrees of freedom [1, 2, ...

The resources required to characterize the dynamics of engineered quantum systems-such as quantum computers and quantum sensors-grow exponentially with system size. Here we adapt techniques from compressive sensing to exponentially reduce the experimental configurations required for quantum process tomography. Our method is applicable to processes that are nearly sparse in a certain basis and can be implemented using only single-body preparations and measurements. We perform efficient, highfidelity estimation of process matrices of a photonic two-qubit logic gate. The database is obtained under various decoherence strengths. Our technique is both accurate and noise robust, thus removing a key roadblock to the development and scaling of quantum technologies. Understanding and controlling the world at the nanoscale-be it in biological, chemical or physical phenomena-requires quantum mechanics. It is therefore essential to characterize and monitor realistic complex quantum systems that inevitably interact with typically uncontrollable environments. One of the most general descriptions of the dynamics of an open quantum system is a quantum map-typically represented by a process matrix [1]. Methods to identify this matrix are collectively known as quantum process tomography (QPT) [1,2]. For a d-dimensional quantum system, they require Oðd 4 Þ experimental configurations: combinations of input states, on which the process acts, and a set of output observables. For a system of n qubits-two level quantum systemsd ¼ 2 n . The required physical resources hence scale exponentially with system size. Recently, a number of alternative methods have been developed for efficient and selective estimation of quantum processes [3]. However, full characterization of quantum dynamics of comparably small systems, such as an 8-qubit ion trap [4], would still require over a billion experimental configurations, clearly impractical. So far, process tomography has therefore been limited by experimental and off-line computational resources, to systems of 2 and 3 qubits [5][6][7].Here we adapt techniques from compressive sensing to develop an experimentally efficient method for QPT. It requires only Oðs logdÞ configurations if the process matrix is s compressible in some known basis, i.e., it is nearly sparse in that it can be well approximated by an s-sparse process matrix. This is usually the case, because engineered quantum systems aim to implement a unitary process which is maximally sparse in its eigenbasis. In practice, as observed in liquid-state NMR [8], photonics [5,9,10], ion traps [11], and superconducting circuits [6], a near-unitary process will still be nearly sparse in this basis, and still compressible. The near sparsity is due to few dominant system environment interactions. This is more apparent for weakly decohering systems [12].We experimentally demonstrate our algorithm by estimating the 240 real parameters of the process matrix of a canonical photonic two-qubit gate, Fig. 1, from a reduced number of configurations. From...

We use Pontryagin's minimum principle to optimize variational quantum algorithms. We show that for a fixed computation time, the optimal evolution has a bang-bang (square pulse) form, both for closed and open quantum systems with Markovian decoherence. Our findings support the choice of evolution ansatz in the recently proposed Quantum Approximate Optimization Algorithm. Focusing on the Sherrington-Kirkpatrick spin-glass as an example, we find a system-size independent distribution of the duration of pulses, with characteristic time scale set by the inverse of the coupling constants in the Hamiltonian. The optimality of the bang-bang protocols and the characteristic time scale of the pulses provide an efficient parameterization of the protocol and inform the search for effective hybrid (classical and quantum) schemes for tackling combinatorial optimization problems. For the particular systems we study, we find numerically that the optimal nonadiabatic bang-bang protocols outperform conventional quantum annealing in the presence of weak white additive external noise and weak coupling to a thermal bath modeled with the Redfield master equation.

Complete positivity of quantum dynamics is often viewed as a litmus test for physicality, yet it is well known that correlated initial states need not give rise to completely positive evolutions. This observation spurred numerous investigations over the past two decades attempting to identify necessary and sufficient conditions for complete positivity. Here we describe a complete and consistent mathematical framework for the discussion and analysis of complete positivity for correlated initial states of open quantum systems. This formalism is built upon a few simple axioms and is sufficiently general to contain all prior methodologies going back to Pechakas, PRL (1994). The key observation is that initial system-bath states with the same reduced state on the system must evolve under all admissible unitary operators to system-bath states with the same reduced state on the system, in order to ensure that the induced dynamical maps on the system are well-defined. Once this consistency condition is imposed, related concepts like the assignment map and the dynamical maps are uniquely defined. In general, the dynamical maps may not be applied to arbitrary system states, but only to those in an appropriately defined physical domain. We show that the constrained nature of the problem gives rise to not one but three inequivalent types of complete positivity. Using this framework we elucidate the limitations of recent attempts to provide conditions for complete positivity using quantum discord and the quantum data-processing inequality. The problem remains open, and may require fresh perspectives and new mathematical tools. The formalism presented herein may be one step in that direction.Comment: 15 pages, 8 figure

By analyzing the dissipative dynamics of a tunable gap flux qubit, we extract both sides of its two-sided environmental flux noise spectral density over a range of frequencies around 2kBT /h ≈ 1 GHz, allowing for the observation of a classical-quantum crossover. Below the crossover point, the symmetric noise component follows a 1/f power law that matches the magnitude of the 1/f noise near 1 Hz. The antisymmetric component displays a 1/T dependence below 100 mK, providing dynamical evidence for a paramagnetic environment. Extrapolating the two-sided spectrum predicts the linewidth and reorganization energy of incoherent resonant tunneling between flux qubit wells.

Two long-standing open problems in quantum theory are to characterize the class of initial system-bath states for which quantum dynamics is equivalent to (i) a map between the initial and final system states, and (ii) a completely positive (CP) map. The CP map problem is especially important, due to the widespread use of such maps in quantum information processing and open quantum systems theory. Here we settle both these questions by showing that the answer to the first is "all", with the resulting map being Hermitian, and that the answer to the second is that CP maps arise exclusively from the class of separable states with vanishing quantum discord.