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We report the first electronic structure calculation performed on a quantum computer without exponentially costly precompilation. We use a programmable array of superconducting qubits to compute the energy surface of molecular hydrogen using two distinct quantum algorithms. First, we experimentally execute the unitary coupled cluster method using the variational quantum eigensolver. Our efficient implementation predicts the correct dissociation energy to within chemical accuracy of the numerically exact result. Second, we experimentally demonstrate the canonical quantum algorithm for chemistry, which consists of Trotterization and quantum phase estimation. We compare the experimental performance of these approaches to show clear evidence that the variational quantum eigensolver is robust to certain errors. This error tolerance inspires hope that variational quantum simulations of classically intractable molecules may be viable in the near future.Comment: 13 pages, 7 figures. This revision is to correct an error in the coefficients of identity in Table

The superposition principle is a fundamental tenet of quantum mechanics. It allows a quantum system to be 'in two places at the same time', because the quantum state of a physical system can simultaneously include measurably different physical states. The preparation and use of such superposed states forms the basis of quantum computation and simulation. The creation of complex superpositions in harmonic systems (such as the motional state of trapped ions, microwave resonators or optical cavities) has presented a significant challenge because it cannot be achieved with classical control signals. Here we demonstrate the preparation and measurement of arbitrary quantum states in an electromagnetic resonator, superposing states with different numbers of photons in a completely controlled and deterministic manner. We synthesize the states using a superconducting phase qubit to phase-coherently pump photons into the resonator, making use of an algorithm that generalizes a previously demonstrated method of generating photon number (Fock) states in a resonator. We completely characterize the resonator quantum state using Wigner tomography, which is equivalent to measuring the resonator's full density matrix.

Nature 454, 310 (2008) Recommended and Commentary by Steven M. Girvin, Yale University Microwaves, despite their name, are particles. However the photon quanta of microwave fields are rather pusillanimous. They carry four to five orders of magnitude less energy than optical photons and are correspondingly vastly more difficult to detect and count. Nevertheless, recent progress in atomic cavity QED [1] and superconducting circuit QED [2] has achieved this. Single-photons-on-demand as well as coherent superpositions of 0 and 1 photons have been generated in a microwave resonator electrical circuit.[3]A classical signal generator produces a sine wave of constant amplitude, frequency and phase. The quantum equivalent (produced by a laser or a microwave signal generator) is a so-called coherent state. Because the phase is sharply defined, the photon number (which is the conjugate variable), is necessarily ill-defined. The number of photons to be found in a coherent pulse is in fact Poisson distributed. As a result, a coherent pulse which contain N photons on average will have a variance in photon number of √N. These closest cousins to classical waves are of course useful but not terribly exciting. There is great current interest in generating highly non-classical states of the electromagnetic field for purposes of quantum communication and quantum information processing. One interesting and highly non-classical class of states are the Fock states. These are electromagnetic pulses which contain exactly n photons where n is some specified integer. Because they have definite photon number, the phase suffers complete quantum uncertainty. Hence the electric field of such pulses is completely uncertain, a fact which has recently been verified. [3] Hofheinz et al. have made a tour-de-force advance by deterministically generating photon number Fock states containing up to N = 6 photons (N = 15 in recent unpublished work) using a superconducting qubit coupled to a resonator.The resonator supports discrete modes at integer multiples of the fundamental. Because the modes are widely spaced in frequency for short res-1

Demonstration of quantum entanglement, a key resource in quantum computation arising from a nonclassical correlation of states, requires complete measurement of all states in varying bases. By using simultaneous measurement and state tomography, we demonstrated entanglement between two solid-state qubits. Single qubit operations and capacitive coupling between two super-conducting phase qubits were used to generate a Bell-type state. Full two-qubit tomography yielded a density matrix showing an entangled state with fidelity up to 87%. Our results demonstrate a high degree of unitary control of the system, indicating that larger implementations are within reach.

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 ...

The simulation of fermionic systems is among the most anticipated applications of quantum computing. We performed several quantum simulations of chemistry with up to one dozen qubits, including modeling the isomerization mechanism of diazene. We also demonstrated error-mitigation strategies based on N-representability that dramatically improve the effective fidelity of our experiments. Our parameterized ansatz circuits realized the Givens rotation approach to noninteracting fermion evolution, which we variationally optimized to prepare the Hartree-Fock wave function. This ubiquitous algorithmic primitive is classically tractable to simulate yet still generates highly entangled states over the computational basis, which allowed us to assess the performance of our hardware and establish a foundation for scaling up correlated quantum chemistry simulations.

Quantized eigenenergies and their associated wave functions provide extensive information for predicting the physics of quantum many-body systems. Using a chain of nine superconducting qubits, we implement a technique for resolving the energy levels of interacting photons. We benchmark this method by capturing the main features of the intricate energy spectrum predicted for two-dimensional electrons in a magnetic field-the Hofstadter butterfly. We introduce disorder to study the statistics of the energy levels of the system as it undergoes the transition from a thermalized to a localized phase. Our work introduces a many-body spectroscopy technique to study quantum phases of matter.