Differentiable Analog Quantum Computing for Optimization and Control
Authors: Jiaqi Leng
Yuxiang Peng
Yi-Ling Qiao
Ming Lin
Xiaodi Wu
Venue: NeurIPS 2022 Accept (2022)
Keywords: analog quantum computing
differentiable programming
auto-differentiation
optimization
quantum control

Abstract: We formulate the first differentiable analog quantum computing framework with specific parameterization design at the analog signal (pulse) level to better exploit near-term quantum devices via variational methods. We further propose a scalable approach to estimate the gradients of quantum dynamics using a forward pass with Monte Carlo sampling, which leads to a quantum stochastic gradient descent algorithm for scalable gradient-based training in our framework. Applying our framework to quantum optimization and control, we observe a significant advantage of differentiable analog quantum computing against SOTAs based on parameterized digital quantum circuits by {\em orders of magnitude}. 

Differentiable Analog Quantum Computing for Optimization and Control
Jiaqi Leng 1.3
Yuxiang Pengi1.2 Yi-Ling 12.4 Ming C. Lin Xiaodi Wu
Joint Center for Quantum Information and Computer Science, University of Maryland DDeparteent of Computer Science, University of Maryland
$ Department of Mathematics, University of Maryland
*Center for Machine Learning, University of Maryland Equal Contribution
Abstract
We formulate the first differentiable analog quantum computing framework with specific parameterization design at the analog signal (pulse) level to better exploit near-term quantum devices via variational methods. We further propose scalable approach to estimate the gradients of quantum dynamics using a forward pass with Monte Carlo sampling, which leads to quantum stochastic gradient descent algorithm for scalable gradient-base based training in our framework. Applying our framework quantum optimization and control, observe a significant advantage differentiable analog quantum computing against SOTAS based on parameterized digital quantum circuits by orders of magnitude.
Introduction
Quantum computing has promised unprecedented improvement in our computational ability to tackle classically intractable problems. Despite of the rapid development of quantum hardware [2 [66] near- term quantum computers are still likely to have very restricted hardware resources, where the limited number of "qubits" and non-negligible machine noises would impede the implementation of large- scale quantum algorithms. Recent research results in both computer science (60) and physics [43] suggest a promising approach of designing resource-efficien Noisy Intermediate-Sen Scale Quantum (NISQ) [51] applications by breaking quantum circuit abstractions and directly designing applications the pulse-level control of quantum machines The benefits of this analog-oriented approach have been witnessed in the history of classical analog computing that predates digital computing due to relaxed hardware requirement and plays important role in domain applications such simulation.
One leading algorithmic paradigm on NISQ computers is the Variational Quantum Algorithm (VQA) with few prominent examples like the Variational Quantum Eignensolver (VQE) (50), quantum approximate optimization algorithm (QAOA) 20), and more in 44. Specifically, VQA uses paramelrized quantum models to characterize loss functions, in particular those from quantum physics, which are potentially intractable for classical computing (26). These parameters will then be optimized, usually through gradient-based approaches, to minimize the given loss function. Conventionally, parameterized quantum models are typically quantum circuits where each gate is parameterized by classical variables. Moreover, auto-differenniaaion techniques have been recently developed (e.g.. [47 55 (77) for a scalable quantum gradient calculation on parameterized quantum circuits. We henceforth refer this existing framework as differentiable digital quantum computing.
Although parameterized quantum circuits are designed for NISQ applications, implementing (digital) quantum circuits still incurs non-negligible overheads, which significantly restricts the size of
Pulse-level control is available IBMQ computers.
36th Conference on Neural Information Processing Systems (NcurIPS 2022).

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parametrized circuits that can be executed faithfully on NISQ machines. Moreover, the current parameterization in VQAs is also largely restricted by available parameterized gates and how they compose circuits, which in turn limits the expressive power of VQAs. A natural alternative to the current digital parameterization to perform VQA directly on parametrized analog signals (pulses), either on digital quantum machines with pulse-level controls, or on general analog quantum hardware. Indeed, parameterized analog pulses have the potential for more efficient NISQ implementation and better expressiveness as suggested in a recent perspective paper [4 which could be a more favorable parameterization for NISQ applications even when digital quantum gates are available.
However, there is not yet systematic study of the analog parameterization for VQAs, its scalable training. and its quantitative benefits in achieving high qualiiyaity solutions in VQA applications.
Contributions. We conduct the first systematic study of the parameteri- zation and scalable training of ana- log VQAs, which we call differen- tiable analog quantum computing. We also leverage our scalable training to demonstrate the quantitative ben- efits of the analog parameterization for specific VQA applications.
DifT. Quantum
Neural ODE
Diff. Physics
Jr x())) =?
(v, (t)
f(x(t), v)
f(x((), v)
Parametrization
Basis Function
Neural Network
Dynamies Equation
Forward
Time Evolution
Forward Pass
Time Integration
Backward
MC Integration
Autodiff
Auto/Manual Diff.
Device
Quantum
Classical
Classical
Machine learning in different dynamical systems. A diagram in Appendixx A illustraaess their connections.
Table I:
Specificallyy, formulate the general differentiable analog quantum computing control problem, miny L(v), where the loss function L(v) is calculated from the v-parametrized quantum state x(T; v) generated by quantum machines at the evolution time t T. Intuitively, this control problem is no different from any classical ones except that the evolution of the quantum state in [0, T" is governed by the Schrödinger equation
dt x(1) = iiH(vv
(1)
where the Hamiltonian operator Hvv, t) can be much more flexibly parameterized in compared with parameterized quantum circuits (detailed in [Section 3.2). and i is the imaginary unit.
We also develop a scalable Monte Carlo integration technique of computing quantum-related gradients from the loss function [(v). A well-known difficulty in computing quantum-relater gradients by classical means is the exponential cost associated with classical simulation of the quantum system. Thus, any scalable solution must leverage quantum machines to compute the gradients for themselves. Existing ditttodifferenttatiin"" techniques for parameterized quantum circuits (c.g.. [47 59 67])
are designed for discrete-time evolution and the basic parameterized units (i.e., gates) are described matrices. Differential analog quantum computing exploits parameterization of continuous-ting evolution and H (v, Ł) refers to a parameterized model from a functional space. Our Monte Carlo integration technique is designed to bridge this technical gap, which is later integrated with the stochastic gradient descent (SGD) for the entire framework for fast convergence and robustness against empirical noise. We further illustrate that our quantum stochastic gradient descent could still work with approximate descriptions of H H(v,) () for domain applications.
by
An analogy could be drawn between approach and classical deep learning, summarized
in Table 1] Neural ODEs [4]] view the structure of ResNet (27]. Xn+11 = Xx + f(n,, 0), as the solving of an ordinary differential equation,
dt x(t) = fxx(t),))
(2)
with f()) as the network layer, the network parameter, and x(t) the hidden state. This formula is similar to our system in [T) although we adopt a different parametrization than neural networks. Similarly, differentiable physics (e.g.. (BI). which incorporates physical simulation into learning process, leverages existing numerical solvers and the autodiff functionality of deep learning with backpropagation to compute gradients of a physical or dynamical system, then integrates them into neural network. This approach has proven to accelerate learning and generalization.
Differentiable analog quantum computing can be deemed as a special form of differentiable physics at the quantum scale. Similar to the promise of differentiable physics or neural ODEs, we have also observed the advantagoous performance of differentiable analog quantum computing against the

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conventional parameterized quantum circuits by orders of magnitude in major VQA applications like optimization (Section 4) and control (Section 5). Our dittreffferentiitionn technique in quantum is however less efficient than classical ones that leverage the chain rule and the forward/backward propagation. This is due to the quantum no-cloning theorem [65] which prevents us from storing any arbitrary immediate state during quantum computation. So, we develop sampling technique to compute gradients. To sum up, the key contributions of this work include:
A formulation of differentiable,
-efficieteff analog quantum computing framework (Sec. A scalable technique of computing gradients by Monte Carlo sampling with SGD (Sec. (.3).
Formal analysis on correctness, efficiency, and robustness that showcases exponential reduction of computational complexity and bounded errors with approximate machine description (Sec. 3.4]].6.)....
Applications of our framework on quantum optimization (Sec. ! and control (Sec. []. with demon- strated advantages by orders of magnitude against parameterized quantum eircuits. Our code is available here: https: / / gi thwo. com/Yil nggiingoooooooooooo di ffquant.m
2 Related work
Learning for dynamics and control. Dynamical systems have widely been used interpret and improve the design of machine learning algorithms. Compared to traditional discrete layers, Neural ODEs [24] demonstrate that continuous modeling of neural network can better learn the continuous structures [22 57 with infinite depth [5] and constant memory cost [70]. Besides neural networks, differentiable physics [6) also computes analytic gradients of classical dynamical systems like rigid
body [14 53), articulated body 23
64 55 soft body [19 21 54], and fluids 662 63 29
61 Those
methods have made great success on design [42] control [28 and system identification
38 tasks in
the macroworld. Our framework, as the first differentiable dynamics for quantum computing, could have tremendous potential chemistry and physies applications.
Quantum machine learning & optimal control. Quantum machine learning is a fast-developing emerging field (e.g., see the survey (5]) where variational quantum algorithms (VQAs) (e.g., see the survey [4] are one of the most promising candidates for NISQ applications. Quantum optimal control (succinetly, quantum control) aims to achieve a desired response from the quantum system by controlling the system parameters 18 6 Quantum control theory has empowered the growth of quantum technologies and found applications in several fields, ranging from molecular chemistry to quantum computing (56) 13.. The connection between quantum control theory and VQAs has been discussed recently [43] 46). Several optimization algorithms have been developed solve quantum control problems, including the Krotov method [4), monotonically convergence algorithms [69], non non-iterative algorithms (68], the GRadient Ascent Pulse Engineering (GRAPE) algorithm 34 the Chopped RAndom- Madis--Baiis/tt (CRAB) algorithm [8) etc. The development of quantum computing also opens up the possibility of solving quantum control problems with quantum computers [T7 9. These proposals take the approach of hybridizing quantum simulations with classical optimization routines. Nevertheless, they either do not support gradient-based methods or compute the gradients in non- scalable way (e.g., via classical simulation), which significantly limits their performance especially comparing to ours. Existing pulse-level variational algorithms [39 46 40 [5] do not discuss their direct application to quantum analog machines and can not compute gradients analytically on quantum machines, while our paper resolves this problem.
3 Differentiable Analog Quantum Computing
Similar to classical dynamical systems, quantum systems also have states, governing equations, and observations, so there naturally exist plenty of optimization (48), control (12), and learning (S] problems for quantum computing. Inspired by the success of gradient-based approach in the classical domain, we propose a differentiable framework compute the gradients of parametrized analog control signals on quantum computers, based on a "forward simulation with stochastic sampling" (see the workflow in Figure
3.1 Quantum preliminaries
qubit (or quantum bit) is the analogue of classical bit in quantum computation. is a two-level quantum-mechantn system described by vectors in the Hilbert space C². We use Dirac notation
3

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Fondback
Lagabaz
Bal:
Vo
H1
Val 0
H
(V.
200 (.00))
Differentiation Analog Quastum Computing
Inotial State
Time Evalution
Fiual State
Figure 1: The workflow of differentiable analog quantum computing. In this 2-qubit example, the system starts with initial ground state au of dimension 4 = 2 and evolves through the time following the Schrödinger equation [T] We control the dynamics of this quantum system by specifying the time-dependent Hamiltonian Hv, f), parameterized by trainable variables In the end of the process, we measure the system and get a real-valued loss value, L. The derivatives computed in the right box. Quantum computers cannot store computational graphs, so propose to sample time in the forward simulation and apply the parameter shift rule to compute the gradients. The derivatives are then used in the feedback loop to update optimizing C.
(Ą5) to denote quantum states (i.e., unit vectors) vb in H. For example, the classical "0" and "1" are represented by [0} = 0 and [1) = [
One qubit states could be in any linear combination of (0) and (1), which is called superposition. An n--ubit state is unit vector in the Kronecker tensor product & of n single-qubit Hilbert spaces, i.c., H. @0 C z C whose dimension is exponential in n. For an by m matrix A and p by q matrix B, their Kronecker product is an np by mg matrix where
(A & Birr-uuqs+u
Bu. The complex conjugate transpose of (αं) is denoted as (+/|= (1)1 (t is the Hermitian conjugate). Therefore, the inner product of & and tb could be written as (ólw).
The time evolution of quantum states under Schödinger equation is specified by the dimeedependent) Hermitian matrix (1) over the corresponding Hilbert space, known as the Hamiltonian of the quantum system. Typical single-qubit Hamiltonians include the famous Pauli matrices:
0 0
1
0
1
0 0
1 0 0 -1
x
Y
Z
=
(3)
Similarly, multi-qubit Hamiltonian can be built from the Pauli group consisting of tensor products of Pauli matrices. Conventionally we write X, (Y,, Zj) for a multi-qubit Hamiltonian to indicate the tensor product of multiple identity matrices while the j-th operand is X (Y, 2), which represents operations on the j-th subsystem.
Quantum measurement refers to the procedure of extracting classical information from quantum systems. It is characterized by an Hermitian matrix M called the observable. Measuring a quantum state (6)) with observable M is modelled random variable whose expectation value is (v²) M (
3.2 Problem formulations
Most computational tasks quantum simulation and control, like finding the ground state of a physics system, can be formulated as the following optimization problem. Given quantum observable M and an initial state (0)) = (b)). we seck for a parameter vector v by minimizing the loss function
L= (W) M (TT(
(4)
where the evolution of (t)) from & = 0 to = T subject to the Schrödinger equation 11) (v,,t) is a Hamiltonian parametrized by v with form
Here,
H(v, () H. + E m
"v,
(5)
where m is the number of control pulses, H. is time- sime-independent Hamiltonian (e.g. Pauli matrices), H, are tensor products of Pauli matrices, and the range of u.(v. () is RR. We also require u, (v, to be differentiable with respect to for any E [0, T). The loss function L. is hence disferentiable. We can loosen the restriction of H, to general time-independent Hamiltonians if we have powerful enough quantum simulators. We keep it in this paper for the convenience of presenting our algorithm.

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With specific M and 10)), optimization problem miny L with post-processing can encode many essential classical and quantum problems. For example, any classical optimization of f(z) over n-bit integers can be formulated as M Σ. [(x) (x)) (a)] More examples like Max-Cut problem and quantum state preparation are discussed in details Sec.
Abstract Quantum Analog Machines (AQAMs). We propose AQAMs as our computational model. An AQAM optimizing the above loss function should be capable of consecutively: (1) evolving under H(v.)) 1) for any v and time interval [4,, 12] C [o, 7); (2) evolving under constant Hamiltonian Hj for time duration r (effectively applying unitary transformation e-HH;T); ); (3) preparing |vo) (4) measuring with observable M. For realistic quantum devices, need to design AQAMs accordingly.
Remark 3.1. A trivial AQAM directly employs the Hamiltonian of a quantum device as H 1). parameterized by tunable pulses. In most realistic quantum devices, multi-qubit interactions not tunable and weak compared to tunable single-qubit Hamiltonians. Thus Hj can be simulated with high fidelity. Our method is robust against imprecise simulations of H (v, t) and H, (discussed in Section 3.6). Hence the trivial AQAMs are usually suitable for realistic quantum devices.
Our formulation of the problem via analog quantum computing is a generalization of the formulation via parameterized circuits. For example, a series of parameterized Pauli rotation gates Rp, (0,) exp(-i(8 ((0(-((0,)2)) can be realized by (v, = Σ, V, (t), with valuation v, = =0,/2, where 1 the indicator function of [ + 1]. However, simulating analog quantum computing via quantum circuits requires much longer time on nowadays devices [43], hence is unrealistic. So direct analog quantum computing can exploit near-term quantum devices much better than quantum circuits.
3.3 Quantum stochastic gradient descent
Our quantum SGD scheme for computing gra- dients on AQAMs is illustrated in this section, with its correctness, efficiency, and robustness discussed in the following sections.
Algorithm 1 Estimating gradients on an AQAM
Classical inputs:
bin, bat (batch sizes), m (number of
control pulses), uy (parametrized pulses), T (evolution duration). v (parameters)
We borrow the idea of mini-batches from classi- cal SGD to deal with the gradients, and set two layers of mini-batches: (1) integration mini- batch with size bii (2) an observation mini- batch with size bobs- The integration mini-batch updates parameters according to the estimation of derivatives on the sampled time. The observa- tion mini-batch repeats experiments to generate more precise measurement results. The scheme is displayed in Algorithm I
Quantum inputs: E)8o (preparation of initial state), H (parametrized system Hamiltonian), H, (pulse- controllable Hamiltonian), EM (measurement procedure with observable M)
Output: a gradient estimation of L to
for E {L, ..., } do
Draw 7 ~ Uniform (0, T)
for j E (1...., m) 8 E {-1, do
for I E {1. bax.] do
The forward and backward propagation of our SGD scheme is depicted in Figure 1] Notice that the inner loop is the only procedure tum machine. The difference of this procedure compared with the estimation of loss function C is the inserted evolution under H₁, which is beneficial in the error analysis in /Section 3.6)
Prepare (&0n) via Eldy)
Evolve under HV, t) for time [0, 7]
Evolve under H, for duration (1 + 8)
quan-
Evolve under Hv, 1) for time [r. T
t- observation sample from Exs
end for Pj (am)))) end for
Bite x! Trdm
With the estimation of gradients, various opti- mizers designed for classical stochastic gradient descent are suitable to optimize the objective function. For example, Adam (35 used in our experiments.
f& 2j=1
Buy (v. T005 - -7)
end for ac dV
T. >kun fr
3.4 Correctness of gradient estimation
We show that Algorithm generates an unbiased estimation of the gradient 0C and hence establishes Zy.
its correctness. The proof of the following theorem is postponed to Aoppppiix B.
5

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Theorem 3.2. The derivative of L to parameters v is
ac T
δv '/" dr E ا-ر j=1 dv (v. (1) (T) (F)
Juj
(6)
(t, (W0) where U. (122, denotes the time evolution operator for time interval [1, t2] under Hamiltonian (v,t).
Here, (1) = (wol (1, ()e ( 1)
One can interpret the above formula as a direct application the chain rule over functional derivatives Duy
&c. and partial derivatives AV since 6Cc Ju, (v,1) () Pj (1) PP (4) by the parameter shift rule 47 59
which is technique for evaluating commutators of Hermitian by quantum processes.
Algorithm esstimates the integral in (6) via Monte Carlo integration (MCT) technique. We prove that the sampling of MCI has finite variances in Appendix B.1 when uy (v,() has bounded derivatives to
Hence MCI converges of rate (b [52]. Other numerical integral methods are also applicable here for different conditions on u and H,. and may have better convergence rates than MCL. We present it here because MCI has good generalization and simplicity. We also remark that similar ideas developed in this section have also appeared in [3] in the circuit model, whereas we develop everything in the analog quantum computing model.
Overall, the forward and backward propagation of differentiation of L are depicted in Figure I Since the parameterization of uj is arbitrary, typical treatment is using Fourier basis or a Legendre basis as the support of the parameterization. Neural networks are also suitable for pulse generation, whose gradients are easy compute via backpropagation.
3.5 Scalability analysis
The resource consumption of our classical-quantmm hybrid approach has two aspects: the classical and the quantum sides. The classical computation, as described in Algorithm 1] has (bimbosssm) numerical calculations. On the quantum side, the sampling process assessing the loss function and its derivatives takes O(T) time each. The total running time on a quantum computer then is Obinbotaaa)) Almost on every architecture of quantum devices, the number control signals m is at most quadratic in the number of qubits (see some survey papers [36] 32 7 (58), showing excellent scalability of our approach. We exhibit the scalability of our method for up to 11 qubits in numerical experiments in /Secieiio 4.2] Our approach could in principle be readily applied the most advanced existing quantum systems (e.g.. 1 with n ~ 60). This is in sharp contrast to the case of GRAPE and CRAB algorithms, which rely classical simulation of quantum systems with exponential cost in n. For instance, the associated classical computation cost for n ~ 60 (i.e. at least 260) prohibitively high, almost reaching the limit of supercomputers today. Our approach makes feasible with only (nn²) complexity.
3.6 Robustness analysis
In this section, analyze systematic errors of gradient estimation produced of Algoooith I Our goal is to optimize the loss function assessed on a realistic and noisy quantum machine, whose capabilities of evolving under H (v.t) and H, are imperfect.
As a concrete example, when using the trivial AQAMs in Remark 3.1 the actual Hamiltonian of the understanding, and the simulation
quantum device may deviate from the description H H(v, t) of H2 may be imprecise due to weak non-tunable terms H.. We show that our algorithm well approximates the gradient of loss function of the actual devices.
One major advantage of Algoorihhm ijis that even though we have imperfect realization of Hamiltonian H (v. t) built in the AQAM, the quantum part is executed the actual quantum machine following the accurate Hamiltonian i (v,,)) !). As result, the output of algorithm well approximates the gradient for the actual quantum machine.
BC Lemma 3.3. Let By denote the accurate gradient of the loss fanction of the quantum machine, ac denote the estimated grudient via Algorithm 1 and ||-Il represents the matrix spectrul norm 130), then
< M|[ maxxe dv (V,
añ 2v
(v,
2v

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The proof of this lemma is detailed in Appendix B.2 In other words, one can use Algorithm 1 to optimize the control pulses without precise understanding of the machine Hamiltonian, the difference Hvv.t) t) Hv, t)) varies slowly w.r.t. v (a rather mild condition to satisfy).
On the contrary, if one relies solely on the Hamiltonian H (v, 6) built in the abstract quantum analog machine (e.g., the classical simulation of quantum systems in GRAPE or CRAB), the approximation error could be as large as the difference H(v, 1) - (v, L) itself, a potentially large term compared with its derivative w.r.t. V. An example is illustrated in Appendix B.3] This particular advantage of Algorithm I exists exactly because of its execution on the real machine, which is potentially applicable in general scenarios where only approximate machine descriptions are obtainable.
Similar to the circuit case, the systematic error caused by the imprecise evolution under Hj is bounded by the error sum in each evolution under H, for duration (1 + 1 )T, which is usually small.
Quantum optimization
Many important optimization problems in both physics and combinatorics that allow variational solutions can be formulated casily in our framework. For example, finding the ground state of physics systems be solved by variational quantum eigensolver (VQE) (e.g. [50 (3]), and searching for the max-cut of graphs can be approximated by quantum approximate optimization algorithms (QAOA) 20).. Replacing parameterized circuits by AQAMs in existing variational quantum algorithms leads to significantly improved convergence based on numerical experiments on classical simulator.
4.1 Variational quantum eigensolver
We exhibit our approach via an AQAM comparable to existing circuit VQE in terms of the pulse duration, with significantly better convergence and hardware efficiency in identifying the ground state of the H2 molecule.
Problem setting. The Hamiltonian of H2 molecule is expanded with Pauli matrices in the form H1, a + an Z Z2 + X2 + Z + az where a, is scalar weight calculated in 33 The ground state v5) has the minimal energy, defined by argmin (g) (xb) Hi,, [+) This energy function is computed on real machines by Pauli measurements and linearity.
Necostm
Upcl
1071
Objective LEmerry
39
Iritsd State
Time Rooletion
Firel Stite
(a) Analog-ansatz-busee VQE for the Ha system
(b) Ground energy search
Fanazak
30
2
0 tpm
(000))
(D101)
4
3
Objective (Cub)
Instal State
Time Evolution
Fizal State
50001
(c) Analog-ansatz-based QAOA for a MaxCut problem
(d) Cut size maximization
Figure 2: Experiments on quantum optimization problems. (a) Experiment results on the H₂ ground energy search. Loss function is the difference of the evaluated energy to the ground energy of H2, and the lower the better. Our method with the same (720dt) or less (360dt) pulse duration converges more than 10 times faster than the circuit model (3 (b) Experiment results finding the max cut for 4 vertices circular graph. Loss function is the difference of the evaluated cut size to the maximal cut size, the lower the better. Our method outperforms the others by orders of magnitude.
7

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AQAM design. We propose an IBM-AQAM mimicking IBM transmon system 44 (a detailed introduction given in Apppnddix q) and hence realizable on IBM's machines. The IBM-AQAM contains 2 qubits and 4 input pulses Uix (t), and can evolve under
H(t) Hsys + E _I Mys(usa,
Here Hys a constant Hamiltonian. The input pulses to IBM quantum devices are (ys(() which are complex values with norm less than 1. These pulses are modulated by the built-in modulation procedure Mis e of the IBM's quantum devices when executed on the real machine. Since the tunable terms have Hamiltonian Xj. we require the IBM-AQAM to be able to evolve under Xj. This is realizable on IBM's machines because in (7) Hay is much weaker than the microwave input pulses for each qubit in form X,, j. ensuring a high fidelity simulation of Hamiltonian Xj on real machines. We also require the 1BM-AQAM to support initializing in state (00) and measuring with M HH and these procedures are easy to implement for IBM's machines. We adopt the parameterization
u5k(v, = [ (VSkio + (v5Rn) Pt (7 (1-1) )
to make the pulse norms less than 1, where .V(0) 0,,V S((=)), for + 0 is differentiable 1te-= is the shifted sigmoid function, T the
normalization function restricting the norm, S(x) = 1-e-"
duration, and Pt is the 1-th Legendre polynomial.
In 33 a parameterized circuit is proposed as layered tunable single qubit rotations and fixed cross- resonance gates, which are compiled to pulses fitting in (7) and sent to the IBM quantum devices. Their one-layer experiments over two qubits have cross resonance gates compiled to pulses with duration around 720dt: where =020.2.... We match it in our experiments, setting T 720dt. Additionally, we test our approach with only half the duration, T = 360dt.
Comparisons. We compare our approach to circuit VQE, finite difference method, simultaneous perturbation stochastic approximation (SPSA), and derivative-fre methods (CMAES (25) and SLSQP 37) with the IBM-AQAM on a classical simulator. The experiment results are displayed in Figure 2b with a detailed hyper-parameter settings in Appendix D.I.I
The circuit VQE converges to L = 0.02, while our approach decreases lower: at 400 epoch, with 720dt it decreases less than 0.002, and with 360dt it decreses 0.01. In general, our approach is over 10 times more accurate than the circuit VQE, and 100 times more accurate than the derivative-fiee methods. The finite difference method and SPSA do not converge because of the intrinsic randomness of quantum measurements at relatively small observation mini-batch (ban. = 100), which would be amplified by the small difference length. With large enough observation mini-batch, the gradients evaluated by finite difference method has ~ 3% relative difference to the gradients evaluated by our approach. These results exhibit the advantages of our differentiable analog framework compared to circuit model and derivative-free analog models
4.2 Quantum approximate optimization algorithm
We also investigate the application of QAOA approximate solutions for the MaxCut problem, an NP NP-complete problem. With a corresponding AQAM, we achieve a significantly better convergence.
Problem setting. Given graph G = {E, V} where V is the vertex set and F = {i.,)) contains all the edges, our goal is to partition the vertices into two sets (Vo, Vi) so that the number of edges between the two sets is maximized. A cut of an n-node graph G is represented by an n-bit string s = by b2. with by E {0, 1} representing in which set the j-th vertex is. We use the computational basis (s) in an n--ubit register to represent the cut s. A maximum cut (s) maximizes the expected cut size (s) C' [ss) where C = 2 Σ (..)) Cj Cj k = Z, Zz. We test the performance on the circular graph shown in the leftmost of Figure 2c
AQAM design. Farhi et al. (20 optimizes p-layer circuit ansatz U(B,7) 7) = IIP - ii where B= Σド= Xy. We set p= 2 as a baseline.
We note that the benefits of using analog controls in VQE for H2, molecule are also discovered in a recent result (46). where the optimization is conducted based GRAPE-like techniques.

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310
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200 200
630
300 400
200
1000
129
(a) (+) preparation
(b) Φ preparation
(c) X gate synthesis
(d) CNOT gate synthesis
Figure 3: Quantum Control. We apply differentiable analog quantum computing framework state preparation and gate synthesis. Four methods (SLSQP, CMAES, GRAPE, and ours) are used in these tasks. In gate synthesis, the accuracy of the built-in gates from IBM Qiskit is shown in dashed lines. Our method outperforms the other three algorithms in terms of convergence rate and accuracy by up orders of magnitude.
To have a fair comparison, we design a Cut-AQAM corresponding to the above circuit: H(t) 1
2 Σ/= (u)o(t)C)5+1 ()))))))).
The input pulses are real functions (t)(t), where we restrict the energy input by requiring |uya| S 1. Cut-AQAM natively supports evolving merely under Cj.+1 Xj by setting djk as indicator functions. We also require it to support initializing in state [0) and measuring with observable C. In our experiment, we set the duration = within which the circuit QAOA can be realized by Cut-AQAM. Similar to (8) we parameterize Uzk (v,t) by normalized linear combination of Legendre's 's polynomial.
Comparisons. The experiments are set up in a similar way as in Section 4.1 with details in Appendix D.1.2] Results shown in FFiuure 2dl The circuit QAOA and SLSQP converges to 0.08 at 200 epoch, while our method converges to 2.6 x 10 Finite difference method, SPSA, and CMAES do not converge to value less than 1. The analysis similar to /Section 4.1 Since Cut-AQAM does not have a high frequency modulation, SLSQP also converges close to 0, but is slower than approach. We conduct larger experiments for up 11 qubits, which shows good scalability and better performance compared to circuit QAOA. For details, see Figure 5 in Appendix D.1.2
Quantum Control
Quantum control problems fall into two categories: 1) state preparation: to steer a given initial state into a target final state; 2) gate synthesis: to effect specific unitary transformation (quantum gate) in the system. In what follows, we discuss how to formulate and solve quantum control problems using our differentiable analog quantum computing framework. We demonstrate methodology by numerical experiments on the IBM-AQAM.
5.1 State Preparation
To prepare the target state |vtar) from certain fixed initial state, we desire parameter vector that minimizes the loss function defined in with the observable M = II- (ttar) (Vtarl. We consider two tasks: 1) to prepare the state |+) from 0): 2) to prepare the two-qubit maximally entangled state ( from (00) In both tasks, the loss function is readily computed as the measurement merely involves local Pauli operators (see Appennix D.2.I), and can be carried out the IBM-AQAM.
In the numerical experiments, the pulse duration is fixed as T 20åt. for the |+) state, and T 1200dt. for the ( state. We also compare our method with two gradient-free methods (SLSQP, CMAES) and the GRAPE algorithm. In both tasks, ours achieves faster convergence than all other three methods. In the [+) task, the final loss value from method reads approximately 10-5, which is 18 times better than the second best result (i.e., SLSQP), as in Figure 3 (a). In the
( task, the final loss value from our method reads 0.034, which is 3 times better than GRAPE and 6 times better than SLSQP, as shown in Figure 3 (b).
V2 (10) (1)) and (++)= (100) [11)).

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5.2 Gate Synthesis
Gate synthesis is more challenging than state preparation because we hope to engineer a full unitary gate Utar instead of just mapping single state to target state. To this end we first specify a set of pairs S = {(x²) ((>x)))l>))){ (y that completely determines Utar in the sense that no unitary map U
other than Utar satisfies (yy) U (xz)
for all j = ..,k. Then, we consider the loss function - = - ビ, Lj. where each 2, is defined as (4) with quantum observable M = I- ly,) (y,l and initial state [xj). When [(v) is close to O, the time evolution controlled by the parameter v is approximately Utar-
The gate and CNOT gate are widely used in digital quantum computing and supported by IBM Qiskit. We now exemplify our method by recovering these two gates on the IBM-AQAM. In both cases, the loss function can be readily computed on IBM machines. See Appendix D.2.2 for detail.
The pulse duration is chosen to be comparable to the built-in ones in IBM Qiskit: T = 160dt for X gate, and T = 1200dt for CNOT gate | to the gate synthesis tasks. The results are shown in Figure 3 (c), (d). In the synthesis of X gate, the final loss value from our method is 1.17 10 which is over 10 times more accurate than the other three methods. In the more involved task of synthesizing CNOT gate, our method returns a pulse sequence with loss 0.0172, while the results from other methods are no less than 0.2. is worth noting that the loss of IBM built-in calibrated pulses evaluated on IBM machines are 0.019 for gate and 0.043 for CNOT gate when no measurement error mitigation or state preparation error mitigation technique is applied.
We apply four methods (SLSQP, CMAES, GRAPE, ours)
Conclusion and Future Work
We have introduced the first differentiable analog quantum computing framework with quantum stochastic gradient descent algorithm that allows directly optimizing the analog pulse control signal on quantum computers. Since the computation history in a quantum system cannot be stored or reused for the purpose of computing gradient, we construct novel formulation for derivatives on quantum computers based on a forward pass with Monte Carlo sampling. With the proposed algorithm, our method outperforms prior methods by orders of magnitude with better hardware efficiency on quantum optimization and control tasks.
Generalization. This work suggests a larger scope of differentiable physics that extends beyond classical cases to quantum scale, where insights and techniques from scale could potentially be translated to the other. For instance, our quantum differentiable framework is largely inspired by the classical counterpart. We also believe some of our findings for quantum computing could also be generally applicable in the classical paradigm: : e.g., our differentiation technique is likely extendable general linear dynamical systems and our robustness analysis also holds in general when only approximate machine descriptions are known.
Limitations and Future Work. As first step, we only describe our framework with a few important, but small, demonstrating examples running on simulators. The great promise of our framework lies in implementing lcage-scale VQA applications on real quantum machines and bringing useful quantum applications to practice. To that end, we plan to incorporate real-world quantum machines into our framework, and solve large-scale VQA tasks.
Acknowledgement
We thank the anonymous reviewers for their helpful feedback. This work was funded in part by U.S. Army Research Office DURIP grant, U.S. Department of Energy, Office of Science, Office of Advanced Scientific Computing Research Award Number DE-SC001 19000099000000 and DE-SC0020273, Air Force Office of Scientific Research under award number FA9550-21-1-0200 U.S. National Science Foundation grant CCF-181669) & CCF-1942837 (CAREER), Barry Mersky and Capital One E-Nnovate Endowed Professorshis.
In IBM Qiskit, the X gate is implemented by a pulse of duration T' = 160dt, and CNOT is by T' 1056dt.
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15


Reviews:
[{"review_id": "nGw5cfNhaf3", "reviewer": "Reviewer_FN4c", "summary": "This work introduces a general technique to evaluate gradients of time-evolved states, in the analog quantum computing setting. ", "questions": "The VQE experiments compare only against finite differences, what about SPSA? ", "limitations": "Yes", "rating": 8, "confidence": 4, "soundness": 4, "presentation": 4, "contribution": 4, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "This is a very interesting and conceptually (very) important work, because it allows to compute exact gradients on large analog quantum computers. While some of the technical tools have been heavily borrowed from ref 3, I believe that the extension to the analog case is crucial. Also, the numerical experiments seem to consistently indicate a superior performance to existing approaches (say, CRAB, etc). \n\n", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "extra_scores": {"ethics_review_area": ""}, "extra_text": {}}, {"review_id": "IqQVqwyw4DW", "reviewer": "Reviewer_cLaN", "summary": "This papers introduces the first differentiable analog quantum computing framework with a quantum SGD algorithm which optimizes the analog pulse control singla on quantum computers.", "questions": "Sorry, I don't know enough in the field to ask relevant questions and initiate fruitful discussions.", "limitations": "No negative societal impact", "rating": 7, "confidence": 1, "soundness": 4, "presentation": 4, "contribution": 4, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "Strengths:\n- The paper looks serious and does what it claims to achieve.\n- The benchmarks are convincing.\n- I liked the fact that there was a robustness analysis.\n- The paper is well written and structured.\n\nWeaknesses:\n- I am not knowledgeable enough in the field to spot weaknesses.", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "extra_scores": {"ethics_review_area": ""}, "extra_text": {}}, {"review_id": "4cvqUvIhcEN", "reviewer": "Reviewer_gTD7", "summary": "The paper studies the problem of variational quantum algorithms and proposes to tune the parameters on the pulse level to achieve better efficiency than optimizations on the gate level. Results on both quantum optimization (QAOA) and quantum control show large advantages. \n", "questions": "1. What are the core differences to the previous pulse-based variational learning methods besides the parameter shift method to obtaining gradients?\n\n2. Figure 1 caption 'an' initial", "limitations": "No significant negative societal impact.", "rating": 5, "confidence": 5, "soundness": 3, "presentation": 3, "contribution": 2, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "Strengths:\n1. Good motivation of optimizations the algorithms on the pulse level to better leverage the current quantum machines. \n2. Good explanations on the parameter shift method to obtain the gradients.\n\nWeekness:\n1. The proposed work is similar to the existing works on pulse-level variational quantum algorithms and quantum optimal control. [1,2,3]\n2. The idea of using parameter shift to obtain gradient is not new. \n3. For parameter shift, the number of circuit runs on real quantum machine is linear to the number of parameters in the circuit so when the size of the circuit is large, the number of gates will growing rapidly. For instance, the pulse length is O(2^(2n)) for quantum optimal control, so the proposed method is not very scalable either.\n\n[1] Meitei, Oinam Romesh, et al. \"Gate-free state preparation for fast variational quantum eigensolver simulations: ctrl-vqe.\" arXiv preprint arXiv:2008.04302 (2020).\n[2] Liang, Zhiding, et al. \"Variational quantum pulse learning.\" arXiv preprint arXiv:2203.17267 (2022).\n[3] de Keijzer, Robert, Oliver Tse, and Servaas Kokkelmans. \"Pulse based Variational Quantum Optimal Control for hybrid quantum computing.\" arXiv preprint arXiv:2202.08908 (2022).", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "extra_scores": {"ethics_review_area": ["I don’t know"]}, "extra_text": {}}, {"review_id": "0ibFSo1WjnG", "reviewer": "Reviewer_vyta", "summary": "The paper proposes a differentiable programming framework for analog quantum computing with a specialized froward scheme based on Monte-Carlo sampling to get estimates of gradients. The rationale of the gradient estimation is justified theoretically. Moreover a sensitivity analysis to modeling errors of the quantum system is provided to handle real applications. Finally, the framework is illustrated on several examples, ranging from quantum optimization and quantum control. In all examples, the proposed framework appears not only efficient but  also outperforms previous methods by some orders of magnitudes. ", "questions": "- One simple, yet powerful addition to the experiments would be to write down exactly the optimization problem (such as min_x f(x) with some constraints) to help the reader understand how quantum computing is used for classical problems and what are the challenges compared to classical optimization. This would greatly help other communities understand how the framework proposed by the authors differ from traditional algorithms. \n- I cannot find any readme file to run the code. I appreciate that the code is commented and a simple readme/tutorial to navigate the code would also be a simple yet great improvement for the paper. \n\n- In the definition of the AQAM: what does evolving under H_j at time t means? \n- For non-experts, it could be good to either provide a quick introduction to the parameter shift rule though reference [50] is good. \n- I do not understand what the authors mean by \"weak\" in \"multi-qubit interactions are not tunable and weak compared to tunable single-qubit Halmitonians\" or in \"may be imprecise due to weak non-tunable terms in H_c\".\n- A reference for the claim \"Almost on every architecture of quantum devices, the number of control signals m is at most quadratic in the number of qubits n\" would be appreciated. \n- For non-experts, it may be good to explain why equation (7) is presented up to Hermitian conjugate terms. \n- A formal statement for equation (7) with assumptions and proof would be appreciated. Generally, a pass on the Appendix from lines 624 to 631 with detailed reasoning would greatly help clarify the theoretical contributions of the paper. \n- Maybe, recall that the dagger symbol is the hermitian operator for non-experts. ", "limitations": "The authors have properly described the limitations of their work, namely the fact that the current experiments are not very large scale although the framework had been designed to be scaled. As the authors say, the current experiments already demonstrate the potential of the framework and the \"limitations\" of the paper can clearly be seen as future work and not dead-ends. ", "rating": 8, "confidence": 3, "soundness": 4, "presentation": 3, "contribution": 4, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "Strengths: \n- The subject of the paper itself, namely proposing a differentiable programming framework for quantum computing is an exciting avenue for research to broaden the applicability of quantum computing. \n- The approach taken by the authors, namely, considering analog quantum computing systems appears clearly driven by a good understanding of the underlying system and is original to me, though I'm not an expert in quantum computing. The authors could explain better the previous approaches. However, their experimental evaluation clearly demonstrates empirically the benefits of their approach. \n- I personally come from a differentiable programming viewpoint, for which the framework posed by the authors exhibits exciting challenges, namely, computing gradient information without access to the intermediate states of the computations. The authors present a simple Monte-Carlo estimator that is easily implemented and already efficient. The framework proposed by the authors can serve as a strong baseline to build upon and provide alternative gradient estimators for analog quantum computing.\n- The authors link their approach to well-known frameworks in differentiable programming such as differentiable physics which help understand the differences and challenges in this setting. \n- Last but not least, the experimental evaluations are well selected and the comparisons show clearly the strength of the proposed approach compared to previous work.  \n\nWeaknesses: \n- While this paper appears to be a strong contribution and an interesting bridge between differentiable programming and quantum computing, it somehow quickly passes over some technical details that could be better explained as asked int he questions section", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "extra_scores": {"ethics_review_area": ""}, "extra_text": {}}]

Author rebuttal:

Decision: Accept