**Authors:** Anthony M. Polloreno

In this work, we bound a machine's ability to learn based on computational limitations implied by physicality. We start by considering the information processing capacity (IPC), a normalized measure of the expected squared error of a collection of signals to a complete basis of functions. We use the IPC to measure the degradation under noise of the performance of reservoir computers, a particular kind of recurrent network, when constrained by physical considerations. First, we show that the IPC is at most a polynomial in the system size \(n\), even when considering the collection of \(2^n\) possible pointwise products of the \(n\) output signals. Next, we argue that this degradation implies that the family of functions represented by the reservoir requires an exponential number of samples to learn in the presence of the reservoir's noise. Finally, we conclude with a discussion of the performance of the same collection of \(2^n\) functions without noise when being used for binary classification.

**Authors:** Anthony M. Polloreno, Kevin C. Young

Coherent errors in quantum operations are ubiquitous. Whether arising from spurious environmental couplings or errors in control fields, such errors can accumulate rapidly and degrade the performance of a quantum circuit significantly more than an average gate fidelity may indicate. As Hastings [1] and Campbell [2] have recently shown, by replacing the deterministic implementation of a quantum gate with a randomized ensemble of implementations, on can dramatically suppress coherent errors. Our work begins by reformulating the results of Hastings and Campbell as a quantum optimal control problem. We then discuss a family of convex programs designed to improve the performance, implementability, and robustness of the resulting mixed quantum gates. Finally, we implement these mixed quantum gates on a superconducting qubit and discuss randomized benchmarking results consistent with a marked reduction in the coherent error. [1] M. B. Hastings, Quantum Information & Computation 17, 488 (2017). [2] E. Campbell, Physical Review A 95, 042306 (2017).

**Authors:** Ariel Shlosberg, Anthony M. Polloreno, Graeme Smith

Quantum error correction is necessary to perform large-scale quantum computations in the presence of noise and decoherence. As a result, several aspects of quantum error correction have already been explored. These have been primarily studies of quantum memory[1, 2], an important first step towards quantum computation, where the objective is to increase the lifetime of the encoded quantum information. Additionally, several works have explored the implementation of logical gates[3-5]. In this work we study a next step - fault-tolerantly implementing quantum circuits. We choose the \([[4, 1, 2]]\) Bacon-Shor subsystem code, which has a particularly simple error-detection circuit. Through both numerics and site-counting arguments, we compute pseudo-thresholds for the Pauli error rate \(p\) in a depolarizing noise model, below which the encoded circuits outperform the unencoded circuits. These pseudo-threshold values are shown to be as high as \(p=3\%\) for short circuits, and \(p=0.6\%\) for circuits of moderate depth. Additionally, we see that multiple rounds of stabilizer measurements give an improvement over performing a single round at the end. This provides a concrete suggestion for a small-scale fault-tolerant demonstration of a quantum algorithm that could be accessible with existing hardware.

**Authors:** Anthony M. Polloreno, Graeme Smith

The Quantum Approximate Optimization Algorithm (QAOA) was originally developed to solve combinatorial optimization problems, but has become a standard for assessing the performance of quantum computers. Fully descriptive benchmarking techniques are often prohibitively expensive for large numbers of qubits (\(n \gtrsim 10\)), so the QAOA often serves in practice as a computational benchmark. The QAOA involves a classical optimization subroutine that attempts to find optimal parameters for a quantum subroutine. Unfortunately, many optimizers used for the QAOA require many shots ($N \gtrsim 1000$) per point in parameter space to get a reliable estimate of the energy being minimized. However, some experimental quantum computing platforms such as neutral atom quantum computers have slow repetition rates, placing unique requirements on the classical optimization subroutine used in the QAOA in these systems. In this paper we investigate the performance of two choices of gradient-free classical optimizer for the QAOA - dual annealing and natural evolution strategies - and demonstrate that optimization is possible even with \(N=1\) and \(n=16\).

**Authors:** Anthony M. Polloreno, Ana Maria Rey, John J. Bollinger

Trapped ions boast long coherence times and excellent gate fidelities, making them a useful platform for quantum information processing. Scaling to larger numbers of ion qubits in RF Paul traps demands great effort. Another technique for trapping ions is via a Penning trap where a 2D crystal of hundreds of ions is formed by controlling the rotation of the ions in the presence of a strong magnetic field. However, the rotation of the ion crystal makes single ion addressability a significant challenge. We propose a protocol that takes advantage of a deformable mirror to introduce AC Stark shift patterns that are static in the rotating frame of the crystal. Through numerical simulations we validate the potential of this protocol to perform high-fidelity single-ion gates in crystalline arrays of hundreds of ions.

**Authors:** Anthony M. Polloreno, Reuben R. W. Wang, Nikolas A. Tezak

In this note we extend the definition of the Information Processing Capacity (IPC) by Dambre et al [1] to include the effects of stochastic reservoir dynamics. We quantify the degradation of the IPC in the presence of this noise. [1] Dambre et al. Scientific Reports 2, 514, (2012)

**Authors:** Anthony M. Polloreno, Arnaud Carignan-Dugas, Jordan Hines, Robin Blume-Kohout, Kevin Young, Timothy Proctor

Randomized benchmarking (RB) protocols are widely used to measure an average error rate for a set of quantum logic gates. However, the standard version of RB is limited because it only benchmarks a processor's native gates indirectly, by using them in composite \(n\)-qubit Clifford gates. Standard RB's reliance on \(n\)-qubit Clifford gates restricts it to the few-qubit regime, because the fidelity of a typical composite \(n\)-qubit Clifford gate decreases rapidly with increasing \(n\). Furthermore, although standard RB is often used to infer the error rate of native gates, by rescaling standard RB's error per Clifford to an error per native gate, this is an unreliable extrapolation. Direct RB is a method that addresses these limitations of standard RB, by directly benchmarking a customizable gate set, such as a processor's native gates. Here we provide a detailed introduction to direct RB, we discuss how to design direct RB experiments, and we present two complementary theories for direct RB. The first of these theories uses the concept of error propagation or scrambling in random circuits to show that direct RB is reliable for gates that experience stochastic Pauli errors. We prove that the direct RB decay is a single exponential, and that the decay rate is equal to the average infidelity of the benchmarked gates, under broad circumstances. This theory shows that group twirling is not required for reliable RB. Our second theory proves that direct RB is reliable for gates that experience general gate-dependent Markovian errors, using similar techniques to contemporary theories for standard RB. Our two theories for direct RB have complementary regimes of applicability, and they provide complementary perspectives on why direct RB works. Together these theories provide comprehensive guarantees on the reliability of direct RB.

**Authors:** Anthony M. Polloreno, Jacob L. Beckey, Joshua Levin, Ariel Shlosberg, James K. Thompson, Michael Foss-Feig, David Hayes, Graeme Smith

We consider estimating the magnitude of a monochromatic AC signal that couples to a two-level sensor. For any detection protocol, the precision achieved depends on the signal's frequency and can be quantified by the quantum Fisher information. To study limitations in broadband sensing, we introduce the integrated quantum Fisher information and derive inequality bounds that embody fundamental tradeoffs in any sensing protocol. These inequalities show that sensitivity in one frequency range must come at a cost of reduced sensitivity elsewhere. For many protocols, including those with small phase accumulation and those consisting of \(\pi\)-pulses, we find the integrated Fisher information scales linearly with \(T\). We also find protocols with substantial phase accumulation can have integrated QFI that grows quadratically with \(T\), which is optimal. These protocols may allow the very rapid detection of a signal with unknown frequency over a very wide bandwidth.

**Authors:** Sabrina S. Hong, Alexander T. Papageorge, Prasahnt Sivarajah, Genya Crossman, Nicolas Didier, Anthony M. Polloreno, Eyob A. Sete, Stefan W. Turkowski, Marcus P. da Silva, Blake R. Johnson

In state-of-the-art quantum computing platforms, including superconducting qubits and trapped ions, imperfections in the 2-qubit entangling gates are the dominant contributions of error to system-wide performance. Recently, a novel 2-qubit parametric gate was proposed and demonstrated with superconducting transmon qubits. This gate is activated through RF modulation of the transmon frequency and can be operated at an amplitude where the performance is first-order insensitive to flux-noise. In this work we experimentally validate the existence of this AC sweet spot and demonstrate its dependence on white noise power from room temperature electronics. With these factors in place, we measure coherence-limited entangling-gate fidelities as high as 99.2 \(\pm\) 0.15%.

**Authors:** C. M. Wilson, J. S. Otterbach, N. Tezak, R. S. Smith, A. M. Polloreno, Peter J. Karalekas, S. Heidel, M. Sohaib Alam, G. E. Crooks, M. P. da Silva

Noisy intermediate-scale quantum computing devices are an exciting platform for the exploration of the power of near-term quantum applications. Performing nontrivial tasks in such devices requires a fundamentally different approach than what would be used on an error-corrected quantum computer. One such approach is to use hybrid algorithms, where problems are reduced to a parameterized quantum circuit that is often optimized in a classical feedback loop. Here we describe one such hybrid algorithm for machine learning tasks by building upon the classical algorithm known as random kitchen sinks. Our technique, called quantum kitchen sinks, uses quantum circuits to nonlinearly transform classical inputs into features that can then be used in a number of machine learning algorithms. We demonstrate the power and flexibility of this proposal by using it to solve binary classification problems for synthetic datasets as well as handwritten digits from the MNIST database. Using the Rigetti quantum virtual machine, we show that small quantum circuits provide significant performance lift over standard linear classical algorithms, reducing classification error rates from 50% to \(<0.1\%\), and from \(4.1\%\) to \(1.4\%\) in these two examples, respectively. Further, we are able to run the MNIST classification problem, using full-sized MNIST images, on a Rigetti quantum processing unit, finding a modest performance lift over the linear baseline.

**Authors:** S. Caldwell, N. Didier, C. A. Ryan, E. A. Sete, A. Hudson, P. Karalekas, R. Manenti, M. Reagor, M. P. da Silva, R. Sinclair, E. Acala, N. Alidoust, J. Angeles, A. Bestwick, M. Block, B. Bloom, A. Bradley, C. Bui, L. Capelluto, R. Chilcott, J. Cordova, G. Crossman, M. Curtis, S. Deshpande, T. El Bouayadi, D. Girshovich, S. Hong, K. Kuang, M. Lenihan, T. Manning, A. Marchenkov, J. Marshall, R. Maydra, Y. Mohan, W. O'Brien, C. Osborn, J. Otterbach, A. Papageorge, J. -P. Paquette, M. Pelstring, A. Polloreno, G. Prawiroatmodjo, V. Rawat, R. Renzas, N. Rubin, D. Russell, M. Rust, D. Scarabelli, M. Scheer, M. Selvanayagam, R. Smith, A. Staley, M. Suska, N. Tezak, D. C. Thompson, T. -W. To, M. Vahidpour, N. Vodrahalli, T. Whyland, K. Yadav, W. Zeng, C. Rigetti

We describe and implement a family of entangling gates activated by radio-frequency flux modulation applied to a tunable transmon that is statically coupled to a neighboring transmon. The effect of this modulation is the resonant exchange of photons directly between levels of the two-transmon system, obviating the need for mediating qubits or resonator modes and allowing for the full utilization of all qubits in a scalable architecture. The resonance condition is selective in both the frequency and amplitude of modulation and thus alleviates frequency crowding. We demonstrate the use of three such resonances to produce entangling gates that enable universal quantum computation: one iSWAP gate and two distinct controlled Z gates. We report interleaved randomized benchmarking results indicating gate error rates of 6% for the iSWAP (duration 135ns) and 9% for the controlled Z gates (durations 175 ns and 270 ns), limited largely by qubit coherence.

**Authors:** M. Reagor, C. B. Osborn, N. Tezak, A. Staley, G. Prawiroatmodjo, M. Scheer, N. Alidoust, E. A. Sete, N. Didier, M. P. da Silva, E. Acala, J. Angeles, A. Bestwick, M. Block, B. Bloom, A. Bradley, C. Bui, S. Caldwell, L. Capelluto, R. Chilcott, J. Cordova, G. Crossman, M. Curtis, S. Deshpande, T. El Bouayadi, D. Girshovich, S. Hong, A. Hudson, P. Karalekas, K. Kuang, M. Lenihan, R. Manenti, T. Manning, J. Marshall, Y. Mohan, W. O'Brien, J. Otterbach, A. Papageorge, J. -P. Paquette, M. Pelstring, A. Polloreno, V. Rawat, C. A. Ryan, R. Renzas, N. Rubin, D. Russell, M. Rust, D. Scarabelli, M. Selvanayagam, R. Sinclair, R. Smith, M. Suska, T. -W. To, M. Vahidpour, N. Vodrahalli, T. Whyland, K. Yadav, W. Zeng, C. T. Rigetti

We show that parametric coupling techniques can be used to generate selective entangling interactions for multi-qubit processors. By inducing coherent population exchange between adjacent qubits under frequency modulation, we implement a universal gateset for a linear array of four superconducting qubits. An average process fidelity of \(\mathcal{F}=93\%\) is estimated for three two-qubit gates via quantum process tomography. We establish the suitability of these techniques for computation by preparing a four-qubit maximally entangled state and comparing the estimated state fidelity against the expected performance of the individual entangling gates. In addition, we prepare an eight-qubit register in all possible bitstring permutations and monitor the fidelity of a two-qubit gate across one pair of these qubits. Across all such permutations, an average fidelity of \(\mathcal{F}=91.6\pm2.6\%\) is observed. These results thus offer a path to a scalable architecture with high selectivity and low crosstalk.