Session M32: Noisy Intermediate Scale Quantum Computers IV

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Partition functions are ubiquitous in physics: they are important in determining the thermodynamic properties of many-body systems, and in understanding their phase transitions. As shown by Lee and Yang, analytically continuing the partition function to the complex plane allows us to obtains its zeros and thus the entire function. Moreover, the scaling and nature of these zeros can elucidate phase transitions. Here we show how to find partition function zeros on noisy intermediate-scale trapped ion quantum computers in a scalable manner, using the XXZ model as a prototype. We illustrate the transition from XY-like behavior to Ising-like behavior as a function of the anisotropy. While quantum computers cannot yet scale to the thermodynamic limit, our work provides a pathway to do so as hardware improves, allowing the determination of critical phenomena for systems that cannot be solved otherwise.   

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Certain aspects of unitary quantum systems are well-described by non-Hermitian effective Hamiltonians, as in the Wigner-Weisskopf (WW) theory for spontaneous decay. Conversely, any non-Hermitian Hamiltonian can be embedded in a unital quantum channel via a generalization of the WW theory. This existence demonstrates the physical relevance of novel features like exceptional points in quantum dynamics and opens avenues for studying many-body systems in the complex plane of couplings. In the case of lattice field theory (LFT), locality/sparsity lends these channels the promise of efficient simulation on standard quantum hardware. We thus consider quantum operations that correspond to Suzuki-Lee-Trotter approximation of LFTs undergoing non-Hermitian dynamics, with potential applicability to spin or gauge models at finite temperature, and/or chemical potential, to quantum phase transitions, and a class of theories with sign problems. We explore these channels on a benchmark, the 1D quantum transverse Ising model with an imaginary longitudinal magnetic field, showing that observables can probe the Lee-Yang edge singularity. Our channels can resolve this despite quantum jumps, and potentially other noise, making a coarse implementation viable for NISQ era technologies   

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The Green's function plays a crucial role when studying the nature of quantum many-body systems, especially strongly correlated systems. Although the development of quantum computers in the near future may enable us to compute energy spectra of classically intractable systems, methods to simulate the Green's function with near-term quantum algorithms have not been proposed yet. Here, we propose two methods to calculate the Green's function of a given Hamiltonian on near-term quantum computers. The first one makes use of a variational dynamics simulation of quantum systems and computes the dynamics of the Green's function in real time directly. The second one utilizes the Lehmann representation of the Green's function and a method which calculates excited states of the Hamiltonian. Both methods require shallow quantum circuits and are compatible with near-term quantum computers. We numerically simulated the Green's function of the Fermi-Hubbard model and demonstrated the validity of our proposals.   

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Synchronization of nonlinear self-sustained oscillators is a wide-spread phenomenon in nonlinear dynamics, engineering, and biology. In the quantum regime, synchronization dynamics shows genuine quantum features, which stem from energy quantization and interference phenomena. Observing these effects experimentally is challenging because both coherent and dissipative dynamics of a nonlinear oscillator must be controlled in the quantum regime. However, recently, we demonstrated quantum synchronization effects by a digital quantum simulation of a spin-1 limit-cycle oscillator on the IBM Q system. In this talk, we will discuss how the perturbative structure of quantum synchronization dynamics can be used to simplify the well-known algorithm of digital quantum simulation. This modification renders the circuit compatible with current noisy intermediate-scale quantum hardware while it ensures that one can still observe quantum synchronization effects.   

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Compared to fault-tolerant quantum computing, variational quantum algorithms (VQAs) hold the hope for a quantum advantage of practical relevance in a not too distant future. However, optimization of circuit parameters remains arduous and is impeded by many obstacles such as the presence of barren plateaus and many local minima in the optimization landscape. Hence, developing more efficient strategies for training parametrized quantum circuits (PQCs) is needed to unlock the full potential offered by VQAs. Extending ideas from the field of meta-learning, we address this task from an initialization perspective, and propose a FLexible Initializer for Parametrized quantum circuits (FLIP) scheme which can be applied to any family of PQCs. FLIP is tailored to learn the structure of successful parameters from a small number of related problems used as the training set. Once trained it can be used on similar problems and show a dramatic advantage over random initialization and also over more involved meta-learning initialization strategies. Furthermore FLIP accommodates quantum circuits of arbitrary sizes and we show that it can be employed on circuits larger than the ones seen during training: a feature lacking in other meta-learning parameter initializing strategies proposed to date.   

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Modern quantum optimization methods feature quantum circuits with partially-ordered modular levels, such as the Quantum Alternating Operator Ansatz¹. For QAOA to work in practice on NISQ devices, it must extract value from quantum effects despite daunting fidelity degradation due to noise². Our experiments on a Rigetti processor, featuring these kind of algorithms, employ calibrated parametric CZ³ and XY^4,5 gates, as well as analog-controlled phases between qubit pairs programmed with a low-level pulse design language (Quilt). We show experimental benchmark results on a 32-qubit chip for circuits related to hard-constrained scheduling problems as well as MaxCut.

¹Hadfield et al. 2019. From the quantum approximate optimization algorithm to a quantum alternating operator ansatz. Algorithms 12(2)
²Marshall et al. 2020. Characterizing local noise in QAOA circuits. arXiv:2002.11682
³Caldwell et al. 2018. Parametrically activated entangling gates using transmon qubits. PRApplied 10(3):034050
⁴Abrams et al. 2019. Implementation of the XY interaction family with calibration of a single pulse. arXiv:1912.04424
⁵Wang et al. 2020. XY mixers: Analytical and numerical results for the quantum alternating operator ansatz. PRA 101(1), 012320   

Live

The variational quantum eigensolver (VQE) is a promising technique for the near-term application of quantum coherent processors to the study of chemical and physical systems. However, the maximum accessible problem size is limited by the number of qubits in a processor that can be mutually entangled with high fidelity. For instance, swap operations needed to distribute entanglement across a device with limited connectivity can quickly consume the available error budget. Here we discuss how to soften this limitation by leveraging the structure of certain simulation problems to reduce the required circuit size. In the first half of our two-part presentation, we will discuss the theoretical formulation of our approach, and applicability to problems of interest. In the second half, we will present data from quantum hardware demonstrating the viability of the approach, and discuss technical details of the experimental realization.   

Live

The variational quantum eigensolver (VQE) is a promising technique for the near-term application of quantum coherent processors to the study of chemical and physical systems. However, the maximum accessible problem size is limited by the number of qubits in a processor that can be mutually entangled with high fidelity. For instance, swap operations needed to distribute entanglement across a device with limited connectivity can quickly consume the available error budget. Here we discuss how to soften this limitation by leveraging the structure of certain simulation problems to reduce the required circuit size. In the first half of our two-part presentation, we will discuss the theoretical formulation of our approach, and applicability to problems of interest. In the second half, we will present data from quantum hardware demonstrating the viability of the approach, and discuss technical details of the experimental realization.   

Live

A significant problem for existing quantum computers is noise. While there are many distinct noise channels, the depolarizing noise model can often appropriately describe average noise for large circuits involving many qubits and gates. We present a method to mitigate the global depolarizing noise by first estimating its rate by executing a noise estimation circuit and then correcting the output of the target circuit using the estimated noise rate. A noise estimation circuit is a circuit that has a similar structure as the target circuit, has a well-defined output, and is classically simulable. We further utilize randomized compiling to convert incoherent noise to coherent noise. The method is experimentally validated on several test cases.   

Live

Achieving near-term quantum advantage will require accurate estimation of quantum observables despite significant hardware noise. For this purpose, we propose a novel, scalable error-mitigation method that applies to gate-based quantum computers [1]. The method generates training data {X_i^noisy, X_i^exact } via quantum circuits composed largely of Clifford gates, which can be efficiently simulated classically, where X_i^noisy and X_i^exact are noisy and noiseless observables respectively. Fitting a linear ansatz to this data then allows for the prediction of noise-free observables for arbitrary circuits. We analyze the performance of our method versus the number of qubits, circuit depth, and number of non-Clifford gates. We obtain an order-of-magnitude error reduction for a ground-state energy problem on 16 qubits in an IBMQ quantum computer and on a 64-qubit noisy simulator.
[1] P. Czarnik, A. Arrasmith, P. J. Coles, L. Cincio, arXiv:2005.10189.   

Live

Correlated noise such as the 1/f noise found in many solid-state systems limits the improvement of gate fidelities in quantum information processors because it is unfeasibly expensive to simulate using conventional methods. This issue is exacerbated when considering algorithms in the presence of noise with correlation times larger than gate durations as gate errors may interfer. Previously, we have shown how filter functions can be used to compute the effect of such noise on gate sequences. Here, I will show how the effect can be efficiently mitigated in gradient-based robust quantum control by introducing analytical results for the derivatives of filter functions. Filter functions represent a spectrally resolved susceptibility to external perturbations and allow for computing complete quantum processes and hence also gate fidelities or leakage rates. Using the analytical expressions for the gradients, we can efficiently optimize control pulses for high-fidelity gates. As an example, I show how one can obtain Rabi pulses that are robust against 1/f noise. I conclude by presenting the open-source filter_functions software framework (https://github.com/qutech/filter_functions), which facilitates computing filter functions and their derivatives for arbitrary quantum gates.   

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Can alternative approaches to phase estimation yield better results for chemical simulation on NISQ hardware? Robust phase estimation (RPE) calculates the difference in phases between two eigenstates of a unitary, provided oracular access to that unitary and the relevant eigenstates. In contrast to conventional phase estimation, it does not require a controlled-U and is naturally resistant to state preparation and measurement errors.

We evaluate the suitability of RPE on near-term hardware. In particular, we calculate the energy landscapes of H₂ and H₃ on extant quantum hardware (pre-compiling to emulate hardware with significantly higher fidelities). Resource requirements for larger molecules are considered.   

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State tomography and other measures of the global properties of a quantum state are indispensable tools in understanding many body physics through quantum simulators. Unfortunately, the number of measurements of the system required to estimate these global quantities scales exponentially with system size. The use of random rotations can greatly reduce the basis of this exponent. We focus on experimentally relevant protocols for state tomography and state purity measurement in superconducting qubits, relying on simple X/Y rotations only. We show a framework for evaluating such protocols, and analytically calculate the required number of measurements, finding significant improvement in the scaling behavior.   

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Achieving near-term quantum advantage will require effective methods for mitigating hardware noise. Data-driven approaches to error mitigation are promising, with popular examples including zero-noise extrapolation (ZNE) and Clifford data regression (CDR). Here we propose a novel, scalable error mitigation method that conceptually unifies ZNE and CDR. Our approach, called variable-noise Clifford data regression (vnCDR), significantly outperforms these individual methods in numerical benchmarks. vnCDR generates training data first via near-Clifford circuits (which are classically simulable) and second by varying the noise levels in these circuits.
We test out new methods by employing a noise model obtained from IBM's Ourense quantum computer to benchmark our method. For the problem of estimating the energy of an 8-qubit Ising model system, vnCDR improves the absolute energy error by a factor of 33 over the unmitigated results and by factors 20 and 1.8 over ZNE and CDR, respectively. For the problem of correcting observables from random quantum circuits with 64 qubits, vnCDR improves the error by factors of 2.7 and 1.5 over ZNE and CDR, respectively.   

Live

We present a variation of the randomized benchmarking (RB) protocol in which random sequences are drawn from a twirling group containing entangling gates with arbitrary rotation angles. Under standard RB assumptions, the average noise channel twirls to a sum of projectors onto the irreducible representations (irreps) of the group. The decay parameters associated with the irreps are isolated by measuring a suitable POVM and applying a transformation to the outcome distribution. We apply this technique to several groups, including the group generated by the two-qubit Pauli operators and continuously parametrized Molmer-Sorenson (MS) gates, which are realizable as the native entangling gates in a trapped ion architecture. The protocol estimates the average fidelity and determines the block-diagonal components of the noise on the MS gates projected onto each irrep. We simulate this method and show that the precision of the noise estimates compares favorably to joint process and measurement tomography, indicating that the tomographic information can be extracted in a manner that is robust to state prep and measurement errors.