We present a quantum algorithm for systems of (possibly inhomogeneous) linear ordinary differential equations with constant coefficients. The algorithm produces a quantum state that is proportional to the solution at a desired final time. The complexity of the algorithm is polynomial in the logarithm of the inverse error, an exponential improvement over previous quantum algorithms for this problem. Our result builds upon recent advances in quantum linear systems algorithms by encoding the simulation into a sparse, well-conditioned linear system that approximates evolution according to the propagator using a Taylor series. Unlike with finite difference methods, our approach does not require additional hypotheses to ensure numerical stability.

The circuit model of a quantum computer consists of sequences of gate operations between quantum bits (qubits), drawn from a universal family of discrete operations [1]. The ability to execute parallel entangling quantum gates offers clear efficiency gains in numerous quantum circuits [2][3][4] as well as for entire algorithms such as Shor's factoring algorithm [5] and quantum simulations [6,7]. In cases such as full adders and multiplecontrol Toffoli gates, parallelism can provide an exponential improvement in overall execution time. More importantly, quantum gate parallelism is essential for the practical fault-tolerant error correction of qubits that suffer from idle errors [8,9]. The implementation of parallel quantum gates is complicated by potential crosstalk, especially between qubits fully connected by a commonmode bus, such as in Coulomb-coupled trapped atomic ions [10,11] or cavity-coupled superconducting transmons [12]. Here, we present the first experimental results for parallel 2-qubit entangling gates in an array of fullyconnected trapped ion qubits. We demonstrate an application of this capability by performing a 1-bit full addition operation on a quantum computer using a depth-4 quantum circuit [4,13,14]. These results exploit the power of highly connected qubit systems through classical control techniques, and provide an advance toward speeding up quantum circuits and achieving fault tolerance with trapped ion quantum computers.Trapped atomic ions are among the most advanced qubit platforms [10,11], with atomic clock precision and the ability to perform gates in a fully-connected and reconfigurable qubit network [15]. The high connectivity between trapped ion qubits [16] is mediated by optical forces on their collective motion [17][18][19][20], and can be scaled in a modular fashion using a variety of methods [10,11]. Within a single large chain of ions, gates can be performed by appropriately shaping the laser pulses that drive select ions within the chain. Here, the target qubits become entangled through their Coulomb-coupled motion, and the laser pulse is modulated such that the motional modes are disentangled from the qubits at the end of the operation [21][22][23]. The execution of multiple parallel gates in this way requires more complex pulse shapes, not only to disentangle the motion but also to entangle exclusively the intended qubit pairs. We achieve this type of parallel operation by designing appropriate optical pulses using nonlinear optimization techniques.We perform parallel gates on a chain of five atomic 171 Yb + ions, with resonant laser radiation used to lasercool, initialize, and measure the qubits. Coherent quantum gate operations are achieved by applying counterpropagating Raman beams from a single mode-locked laser, which form beat notes near the qubit difference frequency. Single-qubit gates are generated by tuning the Raman beatnote to the qubit frequency splitting ω 0 and driving resonant Rabi rotations (r gates) of defined phase and duration. Two-qubit (xx) gates are re...

Product formulas can be used to simulate Hamiltonian dynamics on a quantum computer by approximating the exponential of a sum of operators by a product of exponentials of the individual summands. This approach is both straightforward and surprisingly efficient. We show that by simply randomizing how the summands are ordered, one can prove stronger bounds on the quality of approximation for product formulas of any given order, and thereby give more efficient simulations. Indeed, we show that these bounds can be asymptotically better than previous bounds that exploit commutation between the summands, despite using much less information about the structure of the Hamiltonian. Numerical evidence suggests that the randomized approach has better empirical performance as well.