No abstract

As physical implementations of quantum architectures emerge, it is increasingly important to consider the cost of algorithms for practical connectivities between qubits. We show that by using an arrangement of gates that we term the fermionic swap network, we can simulate a Trotter step of the electronic structure Hamiltonian in exactly N depth and with N^{2}/2 two-qubit entangling gates, and prepare arbitrary Slater determinants in at most N/2 depth, all assuming only a minimal, linearly connected architecture. We conjecture that no explicit Trotter step of the electronic structure Hamiltonian is possible with fewer entangling gates, even with arbitrary connectivities. These results represent significant practical improvements on the cost of most Trotter-based algorithms for both variational and phase-estimation-based simulation of quantum chemistry.

We construct quantum circuits which exactly encode the spectra of correlated electron models up to errors from rotation synthesis. By invoking these circuits as oracles within the recently introduced "qubitization" framework, one can use quantum phase estimation to sample states in the Hamiltonian eigenbasis with optimal query complexity O(λ/ ) where λ is an absolute sum of Hamiltonian coefficients and is target precision. For both the Hubbard model and electronic structure Hamiltonian in a second quantized basis diagonalizing the Coulomb operator, our circuits have T gate complexity O(N + log(1/ )) where N is number of orbitals in the basis. This enables sampling in the eigenbasis of electronic structure Hamiltonians with T complexity O(N 3 / + N 2 log(1/ )/ ). Compared to prior approaches, our algorithms are asymptotically more efficient in gate complexity and require fewer T gates near the classically intractable regime. Compiling to surface code faulttolerant gates and assuming per gate error rates of one part in a thousand reveals that one can error correct phase estimation on interesting instances of these problems beyond the current capabilities of classical methods using only about a million superconducting qubits in a matter of hours.

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.

Quantum simulation of chemistry and materials is predicted to be an important application for both near-term and fault-tolerant quantum devices. However, at present, developing and studying algorithms for these problems can be difficult due to the prohibitive amount of domain knowledge required in both the area of chemistry and quantum algorithms. To help bridge this gap and open the field to more researchers, we have developed the OpenFermion software package (www.openfermion.org). OpenFermion is an open-source software library written largely in Python under an Apache 2.0 license, aimed at enabling the simulation of fermionic and bosonic models and quantum chemistry problems on quantum hardware. Beginning with an interface to common electronic structure packages, it simplifies the translation between a molecular specification and a quantum circuit for solving or studying the electronic structure problem on a

We improve the number of T gates needed to perform an n-bit adder from 8n + O (1) [1, 7, 10] to 4n + O(1). We do so via a "temporary logical-AND" construction which uses four T gates to store the logical-AND of two qubits into an ancilla and zero T gates to later erase the ancilla. This construction is equivalent to one by Jones [15], except that our framing makes it clear that the technique is far more widely applicable than previously realized. Temporary logical-ANDs can be applied to integer arithmetic, modular arithmetic, rotation synthesis, the quantum Fourier transform, Shor's algorithm, Grover oracles, and many other circuits. Because T gates dominate the cost of quantum computation based on the surface code, and temporary logical-ANDs are widely applicable, this represents a significant reduction in projected costs of quantum computation. In addition to our n-bit adder, we present an n-bit controlled adder circuit with T-count of 8n + O(1), an out-of-place adder that can be uncomputed without using T gates, and discuss some other constructions whose T-count is improved by the temporary logical-AND.Accepted in Quantum 2018-05-25, click title to verify arXiv:1709.06648v3 [quant-ph]

Superconducting qubits are an attractive platform for quantum computing since they have demonstrated high-fidelity quantum gates and extensibility to modest system sizes. Nonetheless, an outstanding challenge is stabilizing their energy-relaxation times, which can fluctuate unpredictably in frequency and time. Here, we use qubits as spectral and temporal probes of individual two-level-system defects to provide direct evidence that they are responsible for the largest fluctuations. This research lays the foundation for stabilizing qubit performance through calibration, design, and fabrication.

Recent work has dramatically reduced the gate complexity required to quantum simulate chemistry by using linear combinations of unitaries based methods to exploit structure in the plane wave basis Coulomb operator. Here, we show that one can achieve similar scaling even for arbitrary basis sets (which can be hundreds of times more compact than plane waves) by using qubitized quantum walks in a fashion that takes advantage of structure in the Coulomb operator, either by directly exploiting sparseness, or via a low rank tensor factorization. We provide circuits for several variants of our algorithm (which all improve over the scaling of prior methods) including one with O(N 3/2 λ) T complexity, where N is number of orbitals and λ is the 1-norm of the chemistry Hamiltonian. We deploy our algorithms to simulate the FeMoco molecule (relevant to Nitrogen fixation) and obtain circuits requiring about seven hundred times less surface code spacetime volume than prior quantum algorithms for this system, despite us using a larger and more accurate active space.algorithms. Today, the best-scaling quantum algorithms for chemistry in second quantization use plane waves; with either O(N 3 ) gate complexity (with small constant factors) [8, 9] or O(N 2 log N ) gate complexity (with large constant factors and more spatial complexity) [10].A major limitation to using plane waves in second quantization is that one needs a very large number of spin-orbitals to represent many molecular systems to chemical accuracy. The work of [11] suggests resolving this problem by simulating the plane wave Hamiltonian in first quantization to achieve O(N 1/3 η 8/3 ) gate complexity, where η is the number of electrons. With such low scaling in N , one might be able to use an extremely large plane wave basis. Unfortunately, the practicality of that algorithm is unclear because it has not been compiled to explicit circuits, and it is unclear how large the basis would need to be [10].The more obvious remedy to the low resolution of plane waves is to use a more compact basis. Indeed, the majority of proposals for the quantum simulation of chemistry focus on using very compact molecular orbitals. However, using molecular orbitals leads to complex Hamiltonians with coefficients defined in terms of integrals and O(N 4 ) distinct terms. As a consequence, the first quantum algorithms in this representation had gate complexity O(N 11 ) [12, 13]. Since then, a large community of researchers has worked to significantly reduce the cost of simulation in this representation through tighter bounds [13][14][15], better mappings between fermions and qubits [16][17][18][19][20], improved state preparation techniques [21][22][23][24], application of new time-evolution strategies [25][26][27], considerations of fault-tolerant overheads [28][29][30] and other representational and algorithmic insights [31][32][33][34][35][36].The lowest rigorous complexity of prior work on second quantized arbitrary basis chemistry simulation is either the O(N 5 ) scaling of [26], or th...