What is my quantum computer good for? Quantum capability learning with physics-aware neural networks
Authors: Daniel Hothem
Ashe Miller
Timothy Proctor
Venue: NeurIPS 2024 poster (2024)
Keywords: GNN
Quantum Computing
Quantum Benchmarking
Primary area: machine_learning_for_physical_sciences
TL;DR: A new physics-aware neural network architecture is proposed for predicting which quantum circuits a quantum computer will run with high fidelity.

Abstract: Quantum computers have the potential to revolutionize diverse fields, including quantum chemistry, materials science, and machine learning. However, contemporary quantum computers experience errors that often cause quantum programs run on them to fail. Until quantum computers can reliably execute large quantum programs, stakeholders will need fast and reliable methods for assessing a quantum computer’s capability—i.e., the programs it can run and how well it can run them. Previously, off-the-shelf neural network architectures have been used to model quantum computers' capabilities, but with limited success, because these networks fail to learn the complex quantum physics that determines real quantum computers' errors. We address this shortcoming with a new quantum-physics-aware neural network architecture for learning capability models. Our scalable architecture combines aspects of graph neural networks with efficient approximations to the physics of errors in quantum programs. This approach achieves up to $\sim50\%$ reductions in mean absolute error on both experimental and simulated data, over state-of-the-art models based on convolutional neural networks, and scales to devices with 100+ qubits.

What is my quantum computer good for? Quantum capability learning with physics-aware neural networks
Daniel Hothem
Ashe Miller Quantum Performance Laboratory Sandia National Laboratories Albuquerque, NM 87185
Quantum Performance Laboratory Sandia National Laboratories
Livermore, CA 94550 dhothem@sandia,
anmille@sandia,
Timothy Proctor
Quantum Performance Laboratory Sandia National Laboratories Livermore, CA 94550 t tiproct@sandia. gov
Abstract
Quantum computers have the potential to revolutionize diverse fields, including quantum chemistry, materials science, and machine learning. However, contem- porary quantum computers experience errors that often cause quantum programs run on them to fail. Until quantum computers can reliably execute large quantum programs, stakeholders will need fast and reliable methods for assessing quantum computer's 's capability-i.e. the programs it can run and how well it can run them. Previously, off- the-shl neural network architectures have been used model quantum computers' capabilities, but with limited success, because these networks fail to learn the complex quantum physics that determines real quantum computers' errors. We address this shortcoming with new quantum- selysicsswware neural network architecture for learning capability models. Our scalable architecture com- bines aspects of graph neural networks with efficient approximations to the physics of errors in quantum programs. This approach achieves up to 50% reductions mean absolute error on both experimental and simulated data, over state-of state-of-the-art models based on convolutional neural networks, and scales devices with 100+ qubits.
1 Introduction
Quantum computers have the potential to efficiently solve classically intractable problems in quantum chemistry [Cao et al. 2 materials science [Rubin al., (024). machine learning (Harrow et al.) 2009). and cryptography [Shor] 1997). While contemporary quantum computers are approaching the size and noise levels needed solve interesting problems [Arute al. 2019), they are far from being capable of reliably running most useful quantum programs [Proctor et al. 2021a]. Until we build quantum computers capable of executing any and all useful and interesting quantum programs, stakeholders will require fast, reliable, and scalable methods for predicting the programs that a given quantum computer can reliably execute.
The task learning which quantum programs a particular quantum computer can reliably execute is known quantum capability learning (Proctor et al. 2021a]. Quantum capability learning is very
38th Conference on Neural Information Processing Systems (NcurIPS 2024).

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difficult because the number of possible (Markovian) errors plaguing a quantum Computer grows exponentially in its size BBlume-Kohout al. 2022]. i... in the number of qubits (n) it contains. and errors in a quantum program can combine in difficult-topreedice ways [Proctor et al. 2021a]. Most existing approaches to capability learning restrict themselves to learning how well quantum computer executes small set of quantum programs, by running all of those programs and estimating a success metric for each one [Lubinski et al. 2023 Proctor et al. 2024]. While these methods provide insight into quantum computer's s capability, they are not predictive.
Recently, several groups have proposed building predictive models of quantum computer's capability using convolutional neural networks (CNNs) [Elsayed Amer et al. [2022 Hothem et al. 2024b Vadali
al., [024 Hothem al. 2023b) and graph neural networks (GNNs) Wang et al. [002]] However, these neural network-base capability models achieve only modest prediction accuracy when applied
to real quantum computers, because they fail to learn the complex physics that determines real
quantum computers' failures (Hothem et al. 2024b].
neural network (qpa-NNN architecture for quantum capability learning (Fig. D. Our approach uses neural networks with GNN-inspired structures predict the rates of the most physically relevant errors in quantum programs. These predicted error rates are then combined using an efficient approximation to the exact (but exponentially costly) quantum physics formula for how those errors combine to impact a program's 's success rate. Our approach leverages the graph structures that encode the physics of how errors' rates typically depend on both the quantum program being run and how a quantum computer's s qubits are arranged, yet classically tractable task of approximately combining these function. This enables our qpa-NNs
In this work, we introduce a novel quantum-physics--ee
and it offloads the difficult-to-les error rates to predict circuit's performance to an already-knon to vastly outperform the state-of of-the-art CNNs of Hothem et al. (2024b) on both experimental and simulated data without sacrificing the ability to model large devices of 100+ qubits.
Our qpa-NNs are enabled, in part, by focusing on learning a quantum computer's 's capability on high- fidelity quantum programs, which are those programs that a quantum computer correctly executes with high probability. High-fidelity programs are arguably the most interesting programs to study as we care far more about whether a quantum computer successfully executes a program 99% or 90% of the time rather than 1% or 10% of the time.
In a head-to- (MAE) over the CNNs of Hothem et al. (2024b), on average and on the same experimental datasets. Our qpa-NNs achieve average 36% improvement over those CNNs even after fine-tuning those CNNs on the same subset of the training data (high-fidelity programs) used to train our qpa-NNs.
comparison, our qpa-NNs achieve 50% reduction in mean absolute error
Our qpa-NNs' improved performance is likely largely due to their improved ability to model the impact of coherent errors on a program's 's success rate. Off-the-shed networks struggle with coherent errors [Hothem et al.) 2024b), but our qpa-NNs cancel out, making the qpa-NNs much better predictors in the presence of coherent errors. To verify this, we demonstrate that qpa-NNs accurately predict the performance of random circuits run on a hypothetical 4-qubit quantum computer experiencing only coherent errors. Our qpa-NN obtained ~ 50% lower MAE than CNN, averaged across five datasets, and the trained qpa-NN even exhibits moderate performance when making predictions for different class of circuits (random mirror circuits (Proctor et al. 2021a)) simulated on the same hypothetical 4-qubit quantum computer, i.e, our qpa-NNs display moderate prediction accuracy on out-of -uitt--distribution data.
designed to model how these errors add up and
We make the following contributions in our work:
1. We introduce qpa-NNs, a bespoke neural network architecture for modeling the capability of a quantum computer, which outperform state-of-the-art CNN models by 50% on experimental and simulated data.
2. We use our qpa-NNs model the capability of a simulated 100-qubit device; the largest ever neural network capability learning demonstration by a factor of two.
3. We demonstrate, for the first time, how to train NNs predict the process fidelity [Nielsen 2002) of a circuit, which is the most widely used quantum channel error metric.
4. We provide evidence that the improved performance of our qpa-NNs is partly due to their ability to better model the effect of coherent errors, which are known to challenging for other sfate-of-the-an art methods.
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2
Background
In this section, we review the background in quantum computing necessary to understand this paper. See Nielsen and Chuang (0100] for an in-depth introduction to quantum computing and Blumee Kohout al. [2022] Hashim al. [2024] for a thorough description of the errors in quantum computers.
2.1 Quantum computing
A quantum computer performs computations using qubits, which are two-level systems whose pure states are unit vectors complex two-dimensional Hilbert space, H. The pure states of qubits are unit vectors in H The two orthonormal vectors [0) and (1) that are eigenvectors of the Z Pauli operator are identified as the computational basis of H. Errors and noise real quantum computers mean that they are typically in states p that are probabilistic mixtures of pure states.
quantum computation is performed by running quantum program, typically known as quantum circuil (see illustration in Fig. DD. An rt-qubit quantum circuit (c) of depth d is defined by sequence of d layers of logical instructions } Executing c consists of preparing each qubit in [0), applying each L, and then measuring each qubit to obtain an n- n-bit string b. Each layer typically consists of parallel one- and two-qubit gates, and it intended to implement 2 x 2 unitary U UL1). Together, the layers are intended to implement (c) = U (La) ... (L1).
If quantum circuit is implemented without error, its output bit string b is sample from distribution P(c) whose probabilities are given by Pr(h = x) = [xx] (c]]U(c))00 -...).000000000........ 0) where 100-.. 0) = (0) @ @ (0). [æ:) (x)) ... @ (æn)), and x; is the i-th bit in However, when a circuit is executed on a real quantum computer, errors can occur and this means that its output bit string is sample from some other distribution P(c). The process of errors corrupting a quantum computation can be modelled as follows. Each logic layer L; implements the intended unitary superoperator UL,)) U (Li)ptt where p is general n-qubit state, followed by an error channel A, that is completely positive and trace preserving (CPTP) superoperator [Blume-Kohae et al., (022). The imperfect implementation of a circuit is then simply (i(c) = II A, (IL.), and the output bit 0) (00....0)
string b is with probability Pr(b x) TT([=)
2.2 Quantum capability learning
Because quantum computers are eiror-pronc, knowing which quantum circuits particular quantum computer can execute with low error probability is important. Known as quantum capability learning [Proctor al. 2021a Hothem et al. 2024b). this task formally involves learning the mapping between a set of quantum circuits c E C and some success metric s(c) E quantifying how well e runs on quantum computer Q. In this work, stabilizer) circuits [Arronson and Gottesman| 2004), which are sufficient to enable quantum error correction [Campbell al. [017], and two widely-used success metries: probability of successful Irial (PST) a.k.a. success success probability) and the process fidelity (a.k.a. entanglement fidelity) (Hothem
consider a large class of circuits known Clifford (or
al. 2024b Nielsen 2002
PST is defined only for redinitc-outcome circuits, which are circuits whose output distribution has (if without error) support single bit string, b(c). For any such circuit c, PST is defined as
(c) = Pr Pr(measuring b(c) when executing c on Q)-
In practice, PST(c) is estimated by running the cireuit Nahiot > times on and calculating
PST(c) # observations of b(c)
Nahots
Process fidelity is defined for all cireuits, and it quantifies how close the actual quantum evolution of the qubits is to the ideal unitary evolution. It is given by
1 4t Tr [
F(c)
Estimating F(e) is more complicated than estimating PST(c), but efficient methods exist, such as mirror circuit fidelity estimation (Proctor et al. 2022]. Hence, in theory, it possible to efficiently gather training data using either PST(c) F F(c) arbitrarily large quantum computers.

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Figure I: Quantum capability learning with quantum-physies--ware neural networks (qpa- NNs). Our qpa-NNs are novel architecture for learning quantum computer's ' capability, i.c., the mapping from quantum circuits (or programs) how well that imperfect quantum computer can run those circuits. These networks build in physical principles for how errors in quantum circuits occur-which can be expressed in terms of quantum computer's 's connectivity graph- and efficient approximations to the physics of how these errors combine to impact circuit's 's success rate,
2.3 Modelling errors in quantum computers
Our qpa-NNs build in efficient approximations the quantum physics of errors in quantum computers. They do so using the following parameterization of an error channel: A = exp(E, CyG)). Here Gn - } is the of 2 different Hamiltonian (H) and Stochastic (S) elementary error generators introduced by Blume- et al. [2022]. and éj is the rate of error Gj. Not every kind of error process can be represented in this form (e.g., amplitude damping, errors), but this parameterization includes many of the most important kinds of eπтоrs in contemporary quantum computers. Each H and error generator indexed by non-identity element of the n-qubit Pauli group (P",) The Pauli operator indexing an H or S сгтоr indicates the qubits it impacts and its direction, e.g., the H error generator indexed by X & (m(n-)) is coherent error on the 1 qubit and it rotates that qubit around its X axis.
non-Markovian
Our qpa-NNs use approximate formulas for computing PST(c) or F(c) from rates of H and errors, which now review. Consider pushing each error channel A, to the end of the circuit and combining them together, i.e., compute the error channel A(c) defined by l(e)) A(c) U(c).. Then
PST(c) =1
Σ (SP 08)+
PEPX.Y
where sy and 0p the rates of the P-indexed S and H error generators, respectively, in c's error channel A(c), and PX,Y is the set of n-qubit Pauli operators containing at least one X or Y. Similarly,
F(c) 2 - Σ
sp 0g)
PE
Equations 4 and (5) are good approximations for low-error circuits [Madzik et al. 2022] However, they both suffer from the same flaw: they require tracking ((4)) parameters. To address this problem, our qpa-NNs make an approximation: they only account for contributions of polynomially set of errors that contains all those emror which are most likely to be experienced by quantum computer. this work, we chose to account for only local, low-weight errors, i.e., those with initial support small, connected subset of device's connectivity graph.

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3 Neural network architecture
Our neural network architecture (see Fig. D for quantum capability learning combines neural network layers that have GNN-like structures with efficient approximations to the physics of errors in quantum computers. The overall action of our neural networks is map an encoding of circuit a prediction for (c)(c) or F(c). The same network can predict either (C or F(c) by simply toggling between two different output layers that have no trainable parameters. Our architecture is divided into two sequential parts. The first part of our architecture is neural network NN that has the task of learning about the kinds and rates of errors that occur in quantum circuits. We use GNN-like structures within N to embed physics knowledge for how those errors depend on the quantum circuit being run. The second part of our architecture is a function I with no learnable parameters, that turns N's output into a prediction for PCT(c) F(c).
3.1 Physics-aware neural networks for predicting errors in quantum circuits
The neural network N input is a quantum circuit e of depth d(c) represented by (i) tensor T(c) E (0, 10xxd)xd()xma describing the gates in c (see Fig. Ia). and (ii) matrix ((c) E (0. 1 ) deseribing the measurement of the qubits at the end of c. .N maps I(c) to a matrix & E I xgx(((() and
M(c) to a vector m E Rt.. Egj is a prediction for the rate with which error type j occurs during circuit layer i, and m, is prediction for the rate with which error type occurs when measuring the qubits at the end of a circuit. There are 2(4" - 1) different possible error types that can occur in principle (see Section 2) it is infeasible to predict all their rates beyond very small n. However, the overwhelming majority of these errors are implausible, i.e., they are not expected to occur in real quantum computers
Jelume-Kohous et al. 2022]. Our networks therefore predict the rates of every error from a relatively small set of error types G = [G,,..., G33} containing the k most plausible kinds of error. G is a hyperparameter of our networks. It can be chosen reflect the known physics of a particular quantum computer and/or optimized using hyperparameter tuning. In our demonstrations, choose G to contain all one-body H and S errors well as all two-body H and S errors that interact pairs of qubits within h steps on the modelled quantum computer's ' connectivity graph for some constant/ (see Fig. [Ip-c where Fig. D shows an H or S error in G if h > 2). This choice for G encodes the physical principles that errors are primarily either localized to qubit or are two-body interactions between nearby qubits Kohout et al., 2022). The size of G grows with n, and for planar connectivity graphs (as in, e.g., contemporary superconducting qubit systems [Arute et al., ) it grows linearly in n. This results in (n) errors whose rates / must learn to predict.
The internal structures of N are chosen to reflect general physical principles for how E and m depend on c. Eig is a prediction for the rate that Gj occurs in eircuit layer i, and this error corresponds to space/time location within c-because it occurs at layer index or time i and Gj acts on a subset of the qubits Q(G,) (see example in Fig. [o). This error's rate will therefore primarily depend only on the gates in a time- and space-local region around its location in c. Furthermore, this dependence will typically be invariant under time translations (this is true except for some exotic Markovivnn kinds of errors, which we discuss in Section 8.I. We can encode these structures into N by predicting Eij, from a space-tima time "window" of e around the associated error's location using a filter W, that "slides" across the circuit to predict the rate of G2 versus time i. Stated more formally, we predict Ei using a multilayer perceptron N, whereby N, (V, i)) = E,j and ([([()), i) a snippet of [(c) whose temporal origin is i (see Fig. [p) The shape of each filter W, is hyperparameter of our networks and can be designed to reflect general physical principles, the known physics of a particular quantum computing system, and/or optimized with hyperparameter tuning. The particular neural networks we present later herein use filters (c)) i) that snip out only layer and discard the parts of the layer that act on qubits more than 1 steps away from Q(G,) in the quantum computer's 's connectivity graph (e.g., the filter shown in Fig. D corresponds to the error shown in Fig. [T and = 1). This neural network structure has close connections to graph convolution layers [Kipf and Welling 2016 as well as CNNs. We choose this structure as it can model spatially localized crosstalk errors, which are ubiquitous but hard-to-modes class of errors in quantum computers [Sarovar et al. 2020
The network N must also predict the rates of errors that occur during measurements (unless the qpa-NN will only ever predict F F(c) not PST(c)), but these are typically independent of the rates of gate errors (which are predicted by the N2). So we do not use the .V, and their convolutional filters W; to make predictions for m. Instead we use separate but structurally equivalent networks NV with
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corresponding filters WjJ that take M(c) as input and implement only spatial filtering. That is, W simply discards rows from M(c). as, unlike T(c), M(c) has no temporal dimension. The Wi are hyperparameters of our networks allowing us to separately adjust the shape of each W to reflect the known physics of errors induced by measuring qubits. In our demonstrations, our W/ filters have the same structure as the W filters but with an independent l' steps parameter (large I' enables modelling many-qubit measurement crosstalk).
3.2 Processing predicted error rates to predict capabilities
We process ' output to predict (c) or F(c) using a function f with no learnable parameters. This turns A output into the two quantities of interest, and it also makes training / feasible. We cannot easily train .V in isolation because the error matrix & predicted by .N' is not a directly observable quantity. Generating the data needed to train N' directly would require extraordinarily expensive quantum process tomography [Nielsen et al. 2021], which is infeasible except for very small n. In contrast, both PST(c) and F F(c)) can be efficiently estimated (see Section 2 for given circuit
The function f compptes approximation to the value for PST(c) or F(c) predicted by & and m. The matrix & encodes the prediction that c's imperfect action is
( (c)) (a(E))(La)) ... (I)E)((Ta),
where the L; are the d layers of c (see Section and A.(E)
A.(E) is an
error channel parameterized by the 7th column of Equation (6) implies an exact prediction for (ST(c) or F(c) [e.g.. Eq. (3), but exactly computing that prediction involves explicitly creating and multiplying together each of the 4" x 4" matrices in Eq. (6). This is infeasible, except for very small n. Instead our computes efficient approximation to this prediction.
Our function f's action most easily described by embedding into the space of all possible and errors GR resulting in a d x (22n+1 2) matrix & whose columns are k-sparse. However, we never construct these exponentially large matrices. Consider pulling each error channel the end of the circuit, giving (c)) ( ((( where Ay((E) = exp j=
((') ( Gj). Because c contains only Clifford gates and Clifford unitaries preserve the Pauli group [Aaronson and Gottesman 2004), & has columns that are just c-dependent signed permutations of & 's columns. The signed permutations required can be efficiently computed in advance (i.e., as an input encoding step) using an efficient representation of Clifford unitaries [Gidney] [202T]. Furthermore, these permutations can be efficiently represented in two d x k matrices: sign matrix S(c) containing +1 signs to be element wise multiplied with E and a permutation indices matrix P(e) containing integers between and 2 - 2, where P.j specifies what error Gj becomes when pulled through the d - circuit layers after layer i.
exp(E== 2
We now have a representation of E's prediction for the circuit c's error map A(c) as sequence of error maps AT(EE) ... M (E)), and we need to predict PST(c) or F(c). We can do so we can compute E's prediction for the S and H error rates in A(c), as we can then apply Eq. (4) or Eq. (5). To achieve this, we combine the (E()) into a single error map using first-order Baker- Hauudorl (BCH) expansion. Using our embedded representation, this means simply approximating (c) Mc)) R exp(E, "yG')) where 1y Ei=1& (& i.e., we sum of E. To predict F(c) we then simply apply Eq. (4) (meaning summing up v with those elements that correspond to Hamiltonian errors squared). Because measurement errors impact PST(c). to predict (c) we again apply the BCH expansion to combine in the predicted measurement error map exp( cxp(E) E m,G,) and then apply Eq. ( The efficient representation of the overall action of f is illustrated in Fig. (the addition of the measurement error map is not shown).
the rows
Datasets
4.1 Experimental 5-qubit data
We used the 5-qubit datasets from Hothem et al. (2024b) for our experimental demonstrations. Each of these datasets D {(c, (STC)) was gathered by running random and periodic mirror circuits

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(two types of definite- circuits) on 5-qubit IBM Q computers (ibmq london, img essex, i bbtg.buulingeeeoeeebbbbbb burlington, ibmq viio. l purouureeeee and each circuit using Eq. ( Each circuit was run between 1024 and 4096 times, with the exact number depending upon how many times the circuit sampling process generated the circuit (some short, I-qubit circuits were generated multiple times). The random and periodic mirror circuits contained between I and active qubits the circuit's width-and ranged in depth from 3 to 515 layers (alt. 259 layers for the bomg_yorktown dataset).
and estimating the PST of
As we focus on high-PST cireuits, we removed all circuits with PST less than 85% from each dataset, leaving between 864 (ibmq burlington) and 1369 remaining circuits were partitioned into training, validation, and test sets by their original assignment in/hom et al.] (2024b]. This setup enables a direct comparison between our qpa-NNs and the CNNs trained Hothem et al. 2024b). Training set sizes ranged from 682 circuits on i b burlinn ington to 1097 circuits on i imq_ yorktown, with an approximate training, validation, testing split of 80%, 10%, and 10,, respectively.
circuits in each dataset. The
4.2 Simulated 4-qubit data
For our 4-qubit simulations, we generated datasets of 5000 high-fidelity (F(c) 85%) ran- dom circuits, for a hypothetical 4-qubit processor with "ring" geometry (i.e., like that in Fig. [b)). The circuits ranged in width (w) from 1 to qubits, and depth from 1 to 180 cir- cuit layers. We designed each circuit for a randomly chosen subset of w qubits. Each circuit layer was created by i..dd. sampling from all possible circuit layers on the w active qubits. We used a gate set containing two-qubit CNOT gates and 7 different single-qubit gates (specifically (X(T/2) (2), (1T 2), ((3x/2) (3r/22,, X ( (Tr) where P(0) denotes a rotation around the P axis of the Bloch sphere by 0). See Appendix [for additional details.
All circuits were simulated under the same error model, consisting of local coherent (i.e., H) errors, to exactly compute each c's F(c) [Fig. 3 shows a histogram of F(c)]. After removing duplicate circuits, the resulting datasets = (c. F(c))} were partitioned into training, validation, and testing subsets, with a partition of 56.25%. 18.75%. and 25%. respectively. The parameters of the eπror model were randomly selected: each gate was assigned a small error strength, which was then distributed randomly across all possible (local) one- or two-qubit coherent errors, for the one- and two-qubit gates, respectively. We chose a model with only coherent errors as these errors are ubiquitous, they are hard to model accurately and efficiently, and we conjecture that qpa-NNs can model them.
We also generated 5 datasets of 750 random mirror circuits on the same hypothetical 4-qubit quantum computer. Again, the random mirror circuits varied in width from 1 to 4 qubits, and were designed to be run on a randomly selected subset of w qubits. However, instead of i.i.d. sampling of each circuit layer, each circuit was randomly sampled from the class of random mirror circuits on the w qubits. The depth of the mirror circuits ranged from 8 to 174 layers. Because we generated the mirror eircuit datasets to evaluate how well qpa-NNs and CNNs generalize to oft-o--distributio circuits, they were used exclusively as testing sets. To ensure that no training was performed on mirror circuits, we removed any mirror circuits that appeared in the random circuit sets (in actuality, there were no duplicates).
4.3 Simulated 100-qubit data
For our 100-qubit simulation, we generated single dataset of 5000 high-fidelity (F(c) > 91%) random circuits, for a hypothetical 100-qubit quantum computer with a "ring" geometry. All of the circuits had width of 100 qubits, and ranged in depth from to 22 circuit layers. We sampled circuit layers using the same process and gate set as in the 4-qubit simulations.
We simulated every circuit using the same error model, consisting of local, weight-1 S and H errors. As before, the parameters of the error model were randomly selected and the data were partitioned into training, validation, and testing subsets according to 56.25%, 18.75%, 25% split.
Unlike in our 4-qubit simulations, we did not compute F(c) exactly as doing so for a 100-qubit circuit is infeasible in the presence of coherent errors. Instead, we used first-order simulation method to approximate F(c). In this method, F(c) is computed by assigning each gate its own error vector based on the error model, adding up the error vectors layer-wise to compute error vector
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Figure 2: Prediction accuracy on real quantum computers. (a) The mean absolute error of our qpa-NNs (o), the CNNs from Hothem et al. [024b) (o-CNN, +), and fine-tuned CNNs (ft-CNN on the test data. (b) The predictions of the three models for i vigo on the test data, and (c) the distribution of each model's s absolute error the test data, including the 50th 75th 95th and 100th
percentiles (lines) and the means (points).
for each circuit layer, and then computing Fc)) in the second part of a qpa-NN [Figure []] See Appendix C.4 for more details.
4.4 Encoding schemes
We used two different encoding schemes for converting each circuit into tensor. For the CNNs on experimental data, we used the same encoding scheme as |Htthem et al. [2024b]. as we used their data and networks. For all qpa-NNs, and the CNNs simulated data, scheme. As outlined in Section 3] each width-w circuit c is represented by a dineesiimnsional tensor I(c) E [0, 1] [=xd(c) xnca describing the gates e and a matrix (C) E {0, 1)2x describing the measurement of the qubits. The ij-th entry of [(c).
used the following
[c((c) (c)) (c),,
is one-hot encoded vector of what happens to qubit i in layer j. For the hypothetical 4-qubit ring processor, flch 11: channel for each single-qubit gate and four channels for the CNOT gates. There are four CNOT channels to specify if the qubit is the target or control qubit and the interacting qubit is to the left right of qubit i. We used an additional for the experimental data, depending on the quantum computer's geometry. The first row in M (c) the bitstring specifying which qubits are measured at the end of c. When is a definite-outcome circuit, the second row is its target bit string, i.e., the sole bit string in the support of c's outcome distribution when it is executed without error [i.e., P(c)]. Both [cc) and (c) are zero- erro-padded to ensure consistent tensor shape across a dataset.
CNOT channels
Additionally, each circuit e is accompanied by a permutation matrix P(c) E Noxk and sign matrix S(c) [+1]**k, The ij-entry of P(c) specifies which error the j-th tracked error occurring after the i-th layer is transformed into at the end of the circuit. The ij-th entry of S(c) specifies the sign of that error.
5 5-qubit experiments
We now present the results from our head-to head-to-head head comparison between the qpa-NNs and the CNNs on the 5-qubit datasets used in Hothem al. [2024b). [ the mean absolute error (MAE) achieved by the CNNs (+) and the qpa-NNs (o) on each of the datasets. For all datasets, MAE lower for the qpa-NNs than the CNNs, with an average reduction of 50.4% (az = 16.7%. i.e.,
the standard deviation of percent-drop in MAE). The Bayes factor K is between K = 1030 and K 10383 (here, is the ratio of the likelihood of the qpa-NN to the likelihood of the CNN given the test data). This is overwhelming evidence that the gpa-NN is a better model (K > 102 is typically
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Protecion
hoddly 1S1)
Pidcciy 150
Hddiliy 151
Figure 3: Demonstrating our qpa-NNs' accuracy for hard-to-model coherent errors and at scale. (a) Scatter plot of the prediction errors on test data of qpa-NN (o) and CNN (o) trained to predict the fidelity F(c) of random circuits run on a hypothetical 4-qubit quantum computer. The qpa-NN significantly outperforms the CNN. The top subplot contains histogram (green bars) of the ground-truth fidelities.
(b) Prediction errors on out-of-distribution test data, from random mirror circuits. The qpa-NN achieves modest prediction accuracy on this out-of-distribution task, suggesting that the qpa-NNs are accurately learning error rates. (c) Prediction errors on the 100-qubit test data, demonstrating that our qpa-NN approach can accurately predict F(c) for circuits run on large-scale quantum computers.
considered decisive). These results strongly suggest that the extra infrastructure in the qpa-NNs is making a difference.
The improved performance of the qpa-NNs is not because of an increase in model size. For example, the _ondon CNN contains 6, 649, 531 trainable parameters compared to the 1, 218, 348 train- able parameters in the qpa-NN. Moreover, CNNs of similar larger sizes than the qpa-NNs were included in the hyperparameter optimization space of the CNNs [Hothem et al. 2024b].
Nonetheless, comparing the qpa-NNs to the CNNs is somewhat unfair as the CNNs were trained on -iitrr-iiitriittooon circuits- were trained on the entire training dataset from Hothem et al. 2024bb which also contains low-PST circuits. For a fairer comparison, we fine-tuned each CNN (Fig. on the same high-PST training set used to train the qpa-NNs. Fine-tuning typically increased the CNNs' performances (mean 25.1% improvement, 0N 22.3%). However, the qpa- NNs achieve a MAE that is lower than the fine-tuned CNNs by 32.2% on average (ag = 17.3%) and outperform the finettuned CNNs on all six datasets. K is between 1028 and 10238, which is overwhelming evidence that the qpa-NNs are better models than the fine-tuned CNNs.
6 4-qubit simulations
One reason why the extra infrastructure in qpa-NNs may be necessary is that off-the-shel networks struggle with modeling coherent errors [Hothem et al.) 2024b). To test our hypothesis, trained qpa-NN to predict the fidelity F(c) of random circuits executed on hypothetical 4-qubit quantum computer experiencing purely coherent errors. We compared this qpa-NN to hyperparameter--unee CNN trained the same data. Figure shows the results from one representative dataset.
The qpa-NNs again significantly outperform the CNNs. Across the five datasets, the qpa-NNs' averaged a 52.4% reduction in MAE (a,, = 3.00%) on the test data. We also see a significant improvement in the mean Pearson correlation coefficient, Fapa-NN .688 vs. iCNN .749.
We also found that qpa-NNs trained on random circuits are modest predictors of the infidelity of random mirror circuits, which are a different family of circuits. This is an example of out-of- distribution generalization. Random mirror circuits differ in variety of ways from the random circuits on which the qpa-NNs were trained, including both the presence of idle gates (which are noiseless in our simulations) and motion-reverse or cancellation of errors that are far apart in time. The Npa-NNs achieve an average MAE of .72% on the random mirror circuits (o, = .046%). Although this is a 3.2x increase in MAE over the in-distribution test data, the strong linear relation between the network's predictions and the ground truth (f = .912, a. = .009) strongly suggests that the qpa-NNs are learning information relevant to random mirror circuits.
structure in the circuits that causes the addition

---

7 100-qubit simulation
neural networks scale just as well as CNNs, despite their extra infrastructure. To demonstrate their scalability, we trained qpa-NN to predict the fidelity F(c) of random circuits executed on a hypothetical 100-qubit quantum computer experiencing a mix of stochastic and coherent emors. To our knowledge, this is the first creation of capability model of any kind, for a 100+ qubit quantum computer. Figure 3(c) shows the results from our demonstration.
The qpa-NN achieved MAE of 0.097%. While the underlying noise model was quite simple, this result shows that it is technically feasible to construct qpa-NN capability models for today's moderate-scabe quantum computers and for tomorrow's 's early fault-tolcrant quantum computers.
8 Discussion
8.1 Limitations
Our results are significant improvement over the state of the art, but our approach does have several limitations:
1. As presently conceived, approach assumes that the modelled quantum computer's error rates are invariant under time translations, which is a kind of Markovianity assumption (although weaker than the typical Markovian assumption used in conventional quantum computer models [Nielsen et al. 2021]). However, non-Markovian noise exists in quantum computers [White al. 2020). In the future, we plan to address this issue by adding temporal information into our approach, perhaps with a temporal or positional encoding (Vaswani et al. 2017].
2.
Our approach only considers error classes (H and errors). Other Markovian error classes, like amplitude damping, exist, but their error rates e contribute to PST and fidelity at order (E=)) [Mądzik al., 2022]. Our approach can be casily extended include those errors, if necessary, by learning their rates with N and updating f to account for their presence.
3. Our current approach works for Clifford circuits, which includes arguably the most important kinds of circuits (c.g., quantum eπrоr correction circuits) but not all interesting circuits. This is because our method for efficiently propagating errors through circuits (implemented by together with the S and P matrices) leverages the elegant mathematics of Clifford circuits. Our approach can be easily extended to generic few-qubit quantum circuits ( 10 qubits), but to obtain the efficiency needed for large with general circuits we will need to develop approximate methods for propagating errors through those circuits.
8.2 Conclusion
In this paper, we presented a new quantum-physics--av physics-aware neural network architecture for modelling quantum computer's capability that significantly improves upon the state of the art. The new architecture concatenates two parts: (i) neural network with structural similarities to GNNs that uses gate information and a quantum computer's 's connectivity graph to predict the rates of errors in each of circuit's 's layers, and (ii) non-trainable function that turns the predicted error rates into capability prediction. By imbuing these networks with knowledge about how errors oceur and combine within a circuit, we are able to outperform state-of-the-ar CNN-based capability models by ~ 50% on both experimental data and simulated data. We also provided evidence that our guantum-physics- networks are learning the true physical error rates, as they exhibit modest prediction accuracy when predicting the fidelity of cuttof-distribution quantum circuits, which would enable our networks to also be used to diagnose the eиror processes occurring in particular quantum computer (an important task known as characterization or tomography [Nielsen et al. 2021
Understanding which quantum circuits quantum computer can run, and how well it can run them, an important yet challenging component of understanding a quantum computer's power. Given the complexity of the problem, neural networks are likely to play a large role in its solution. As our results demonstrate, our new physics-aware network architecture could play a critical role in building fast and reliable neural network-base capability models.
10

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Acknowledgments and Disclosure of Funding
This material was funded in part by the U.S. Department of Energy, Office of Science, Office of Advanced Scientific Computing Research, Quantum Testbed Pathfinder Program, and by the Laboratory Directed Research and Development program at Sandia National Laboratories. T.P. acknowledges support from an Office of Advanced Scientific Computing Research Early Career Award. We acknowledge the use of IBM Quantum services for this work. The views expressed are those of the authors, and do not reflect the official policy or position of IBM or the IBM Quantum team.
Sandia National Laboratories is multi-mission laboratory managed and operated by National Tech- nology & Engineering Solutions of Sandia, LLC (NTESS), a wholly owned subsidiary of Honeywell International Inc., for the U.S. Department of Energy's 's National Nuclear Security Administration (DOE/NNSA) under contract DE-NA0003525. This written work authored by an employee of NTESS. The employee, not NTESS, owns the right, title and interest in and to the written work and is responsible for its contents. Any subjective views or opinions that might be expressed in the written work do not necessarily represent the views of the U.S. Government. The publisher acknowledges that the U.S. Government retains a non-exclusive, paid-up, irrevocable, world-wide license to publish or reproduce the published form of this written work or allow others to do so, for U.S. Government purposes. The DOE will provide public access to results of federally sponsored research in accordance with the DOE Public Access Plan.
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A
Compute resources
All of the quantum-physics- physics-aware neural networks used in our 5-qubit experiments and 4-qubit simulations were trained using 6-Core Intel Core 19 processor on MacBookPro 15.1 with 32GB of memory. Each model took roughly 15-20 wall clock minutes to train. Total training time, across the paper, totaled ~ 160 wall clock minutes.
All of the 4-qubit simulations and data pre-processing were performed using 6-core Intel Core i9 processor on a MacBookPro 15.1 with 32GB of memory. Each dataset took approximately hour of wall clock time to create. This total includes the initial circuit creation, simulating the circuits, and encoding each circuit into tensor.
All of the 100-qubit simulations, data pre-processing, and model training were performed using two 14-core Intel Xeon CPU E5-2697 v3 @ 2.60GHz processors. An end-to-ene run (i.e., circuit generation to trained model predictions) took roughly 12 hours of wall clock time.
B Code and data availability
The simulated data as well as records of all the quantum physics-aware networks will be released
publicly once they clear Sandia's copyright process. Until then, please email the authors. The CNNs and 5-qubit experimental datasets used in Hothem et al. [2024b] are available at Hothem et al. [2023a]
The datasets were originally located Proctor al. CC-BY 4.0 International license.
2021b). Each dataset was released under
All simulations performed using combination of pygsti version 0.9.11.2 Nielsen al. 2020
and stin version 1.13.0 Gidney 2021]. Models were trained and developed using Keras version 2.10.0 [Chlleet al. 2015) and TensorFlow version 2.12.0 (Abadi et al. 2013]. The physics-aware for
network model classes PST and CircuitErrose
for process fidelity) available in the Supplementary Mate-
rial Hothem et al. 2024a as well as on the feature - ml branch of pygsti.
C Datasets
Device
Geometry types
Circuit
Circuit widths
Circuit depths
Training size
Validation
Test set set size
size
lbmq_london
t-bar
mirror
1-5
3-515
711 cir-
104 cir-
qubits
layers
cuits
cuits
circuits
ibmq_ourense
t-bar mirror
1-5
3-515
930 cir-
124 cir-
114 cir-
qubits
layers
cuits
cuits
cuits
ibmq_essex
t-bar mirror
1-5
3-515
713 cir- 93
86
qubits
layers
cuits
circuits
circuits
Lang_burlington
(-bar mirror
1-5
3-515
682 cir- 90
92
qubits
layers
cuits
circuits
circuits
ibmq_vige vigo
t-bar mirror
1-5
3-515
1029 cir-
137 cir-
126 cir-
qubits
layers
cuits
cuits
cuits
bowtie
mirror
1-5
3-515
1097 cir-
132 cir-
140 cir-
qubits
layers
cuits
cuits cuits
Ring (x5)
ring
random
1-4
1-180
2813 cir-
938 cir- 1250 cir-
i.i.d.
qubits
layers
cuits
cuits
cuits
Ring (x5)
ring
mirror
1-4
8-174
750 cir-
qubits
layers
cuits
100-qubit Ring
ring random
100
1-22 lay-
2812 cir-
938 cir- cuits
1250 cir-
i.i.d.
qubits
ers
cuits
cuits
Table 1: Summary data of every dataset used in the paper. The data for the 4-qubit ring processors averaged over the simulated datasets. See Figure 4 for images of each processor geometry (i.c., the qubit connectivity graph).
14

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We provide additional details on the datasets used in the paper. Table summarizes each dataset. We tracked all weight-2 errors with support on qubits connected by hops in all, but the 100-qubit datasets. Below, we provide additional details on the circuit and emor model generating processes.
C.1 Creating the circuits
(a) bowtie
(c) ring
(b) t-bar
Figure 4: Device geometries. The connectivity graphs for the (a) 5-qubit _ yorktown "bowtie" processor; (b) the remaining 5-qubit experimental "t-bar" processors; and (c) the 4-qubit simulated "ring" processor. The 100-qubit simulated "ring" processor has the same topology, just more qubits.
In this subsection, we go over how the random i.i.d.llyer circuits and random mirror circuits were created for this paper. We start by explaining how we generated the random i.i.d.layer circuits for a 4-qubit ring processor, and then explain the modifications needed to generate the random mirror eircuits. This subsection's content is conceptual. The actual circuits were created in pygsti using the code in the Supplementary Material.
Each random -.id.rlayer circuit e created by multi-step process. First, randomly sampled connected subset Q. S {Q0, Q1, Q2, Q3) of qubits for which e is designed for. Then, we uniformly sampled c's depth from between and du., a pre-determined, circuit- cireuit-width-deee widhht--dhh-dedeen maximum depth. The depths d were selected to ensure that F(c) > 85% given the maximum error strengths used create the simulated error model (Section C.2 Third, we randomly sampled two-qubit gate density P2o between 0 and 2/3. The density p2o determines the average number of two-qubit gates in each of c's layers. We then sampled each layer i.i.d. from all possible circuit layers the qubits in Qe.
The random mirror circuits generated using similar multi-step process with two differences. The first difference is that we used a pre-determined maximum depth of d /../6. We chose to reduce the mre-determined, cireuit-width-den width-dependent maximum depth that the deepest random mirror circuits had roughly the same length as the deepest random i.idd circuits. The second difference is that we created random mirror circuit on Q.. See Proctor et al. 2021a for more details.
C.2 Creating a 4-qubit error model
In this subsection, we explain how we constructed the 4-qubit Markovian local coherent error model used in Section Again, provide conceptual explanation. The actual error model was created in pygsti using the code found in the Supplementary Material.
The 4-qubit Markovian local coherent етrоr model was specified using the error generator framework explained Section g and [Blume-Kohowe et al. [2022]. The error model consists of operation- dependent errors sampled according to two-step process. The eгror strengths for each gate and qubit(() pairs were independently sampled. First, we sampled an overall error strength e, for each one- and two-qubit gate by randomly sampling from j0, 1] and scaling by pre-determined maximum error strength (.025%). Then we sampled the relative error strengths Egt of each of the 4 coherent errors, where 1, for one- and two-qubit gates, respectively. We then normalized Ey.eel
15

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to obtain the actual error strengths according to the following equation:
Eg Vēg g.rel
The re-scaling ensures that, to first order, gate contributes approximately Ey to the circuit's process infidelity (or PST, if appropriate).
C.3 Creating 100-qubit error model
In this subsection, we explain how we constructed the 100-qubit Markovian error model used in Section 7 As with the 4-qubit error model, we provide a conceptual explanation. The actual error model was created in pygsti.
As with the 4-qubit error model, the 100-qubit Markovian error model was specified using the error generator framework explained in Section The error model consisted of gate-dependent errors sampled according to a two-step process. Unlike with the 4-qubit enror model, the 100-qubit error model included non-local Pauli stochastic and coherent errors, and the errors for each gate were independent of the qubit acted upon by the gate. Moreover, all of the errors in the 100-qubit error model are weight-l errors (i.e., they affect single qubit).
Each gate's error strengths were independently sampled. First, we enumerated all 600 possible weight-1 Pauli stochastic and coherent errors in 100-qubit device. Then, for each gate 9, independently sampled the strength of each of the 300 weight-1 Pauli stochastic errors and the 300 possible weight-1 coherent errors. The strengths were sampled uniformly random, with a maximum Pauli stochastic error strength of 0.0000001 and a maximum coherent error strength of 0.00005. The resulting 600 error strengths were assembled into 500-dimensional error vector for the gate, Eg.
C.4 Simulating the 100-qubit circuits
In this subsection, we deseribe the first-order simulation method used to approximate F(e) in our 100-qubit simulations (Section 4.3) The method works by constructing approximate error matrix E(c) [Fig. [)), and then estimating Fcc) by performing the same computation as in the second half of qpa-NN [Fig. []] Herein, describe how to construct the approximate error matrix E(c). Readers should refer to Section 3 for an in-depth explanation on how to use E(c) to estimate F(c).
For each circuit we constructed an approximate eтror matrix E(c) by concatenating approximate error vectors (c) for each circuit layer in e. Each layer's error vector was computed as linear combination of the individual gate enror vectors, with coefficients equal to the number of times each gate appears the circuit layer.
D Networks
D.1
Quantum-physscc--ware network details
Dataset
Metric
Model size
Nhops
N Nerrors
Dense units
5-qubit t-bar
PST(c)
1218348
174
[30, 20, 10, 5,
5,
5-qubit bowtie
PST(c)
1596420
3
210
[30, 20, 10, 5,
5,
4-qubit ring
F(c)
299772
2
132
[30, 20, 10,
5.
100-qubit ring
F(c)
12706200
1
600
[30, 20, 5,
5.
Table 2: Summary data for the quantum-physicc-avve networks used in the paper.
Table 2
briefly outlines the hyperparameters and model sizes of the physics-awan neural networks used in this paper. The Nhops and Nerrors hyperparameters were chosen by hand based upon subject- matter-expert knowledge of the errors in quantum computer. The size and shape of the dense layers
16

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were selected arbitrarily. All dense subunits used a ReLU activation function. All models were trained using keras's Adam [Kingma and Ba 2015) optimizer with a step size of 10 and with mean squared error the loss function. Model training was cut short using early stopping. To help with training, we scaled (c) and F(c) by 10000 when training the physics-aware networks. The notebooks in the Supplementary Material contain more details.
D.2 Convolutional neural network details
Details on the specific convolutional neural networks used in this paper are located in Hothem et al. [2024b). We fine-tuned each network on high-PST experimental data using the Adam optimizer and stopping.
We selected the model architecture for the convolutional neural networks used in our simula- tions via hyperparameter tuning. We performed 100 trials of hyperparameter tuning using the BayesianOptimizer in class in kerastuner. Additional details, including the specific hyperpa- rameter space, are located in the Supplementary Material.
E Experimental results
Dataset
Network
Mean absolute er-
Bayes factor vs. CNN
Bayes factor vs. ft-CNN
ror (%)
qpa-NN
1.09
10340
10
ibmq london
CNN
3.44
ft-CNN
2.39
10156
qpa-NN
1.24
1030
1033
ibmq_ourense
CNN
1.55
ft-CNN
1.68
10 10-
qpa-NN
1.39
10248
10238
ibmq_essee essex
CNN
3.03
ft-CNN
2.82
109.16
qpa-NN
1.25
10152
1038.2
CNN
2.27
ft-CNN
1.61
1093.7
qpa-NN
1.21
10378
10
CNN
2.98
ft-CNN
1.75
10263
qpa-NN
1.19
10383
1028.4
yorktown
CNN
2.71
ft-CNN
1.31
10354
Table 3: Summary model performance data on experimental data.
Table summarizes model performance on each of the experimental datasets used the paper. Copies of the pre-trained quantum physics-aware neural networks, original CNNs, and fine-tuned CNNs are available in the Supplementary Material. Scatter plots of each model's predictions are also available in the Supplementary Material.
quantum-physsic--wvee
F Simulation results
Table summarizes model performance on each of the 4-qubit simulated datasets used in the paper. Copies of the pre-trained caaattu-physic--avae neural networks and fine-tuned CNNs are available in the Supplementary Material. Scatter plots of each model's prediction errors are also available in the Supplementary Material.
17

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Dataset Network
Circuit type
Mean absolute error
Pearson correlation coefficient
(%)
random i.i.d.
.176
.968
qpa-NN
mirror
.720
.914
random i.idd.
.404
.751
CNN
mirror
1.17
.872
random i.id
.200
.600
qpa-NN
mirror
.652
.21
random i.i.d.
.421
.741
CNN
mirror
.922
8611
random i.id.
.190
.970
qpa-NN
mirror
.769
914
2
random i.id.
.406
.732
CNN
mirror
1.20
.833
random i.idd.
.191
.952
qpa-NN
mirror
.719
.912
3
random i.id
.367
.764
CNN
mirror
1.02
.857
random i.i.d.
.195
.960
qpa-NN
mirror
.761
.898
random i.id.
.405
.763
CNN
mirror
1.05
.847
Table Summary model performance data on the 4-qubit simulated data.
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Section[]andge [d
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24


Reviews:
[{"review_id": "XkC5M9qQak", "reviewer": "Reviewer_oCVW", "summary": "The paper presents an approach to improve the state of the art in quantum capability learning, which is the task to predict the prowess for error when running a specific quantum algorithm given a fixed quantum computer. The tackle this, the authors introduce some specializations on (graph) neural networks that fit the nature of quantum computers and their errors especially well. The new approach yields better results than purlely CNN-based methods on a specific synthetic and empirical data.", "questions": "None.", "limitations": "No comments.", "rating": 6, "confidence": 4, "soundness": 2, "presentation": 3, "contribution": 3, "strengths": "The approach is very interesting from both a quantum and a machine learning point of view. The issue of quantum capability learning is described in geat detail and the innovation of the approach becomed clear. In the method to intertwine neural network architectures with their subject of training, especially when that subject is a quantum computer, I see great potential for further investigation.", "weaknesses": "To be frank, I consider the issue of quantum capabilty learning (especially as it is motivated in this paper) a rather artificial one, given the quite small number of potentially useful quantum algorithms. And I think this leads to some rather weak arguments, mostly in the motivation of this paper. However, quantum capability learning (as the authors examine it in this paper) is still an interesting and relevant topic. I suggest cutting down on practical promises and focusing on the pure challenge of predicting a quantum computer's behavior using classical neural networks, which I consider a sufficient motivation for the study of this topic.\n\nWhile the performance-based analysis has good results, I would have wished for a more in-depth take that can actually clarify the speculation posed in the introduction: \"Our qpa-NNs' improved performance is likely largely due to their improved ability to model the impact of coherent errors...\"", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "We are grateful for the reviewer's summary of our paper and their recognition of its strengths. We are also pleased that the reviewer finds modelling a quantum computer's performance with neural networks to be an interesting and worthwhile problem. Your detailed review and constructive feedback will be instrumental in improving our manuscript. Implementing your suggestions will significantly enhance our paper's quality and impact.\nIn response to your feedback, we plan to make the following changes to our paper:\n   - Include new 4-qubit demonstrations on simulated quantum computers experiencing different ratios of coherent to stochastic error rates. \nWe now address, in order, the two listed weaknesses and explain how our new results/edits address each of the weaknesses.\n\n**Weakness 1**: lack of motivation behind quantum capability learning.\n\nWe respectfully disagree with the reviewer’s assertion that the motivation for quantum capability learning is weak given the small number of useful quantum algorithms. We believe that quantum capability learning will be especially important in the early fault-tolerant era as devices grow beyond our abilities to classically simulate, while remaining too noisy or small to reliably implement general quantum algorithms. It is precisely in this early fault-tolerant era when we will need accurate, scalable, and fast-to-query predictive models of a quantum computer’s capability to better understand which experiments to run and which devices to build. Nonetheless, we are happy to read that the reviewer believes that the “pure challenge of predicting a quantum computer’s behavior using classical neural networks” is relevant and sufficiently motivating.\n\n**Weakness 2**: failure to substantiate the claim that the qpa-NNs outperform the CNNs due to their improved ability to model the impact of coherent errors.\n\nWe performed three additional 4-qubit demonstrations to better support our claim that the qpa-NNs improved performance is due to their improved ability to model coherent noise. Each new demonstration used the same setup as the 4-qubit demonstration in the original draft except using a different error model. The error models differed in the ratio of total coherent to stochastic noise allowed in each gate’s error model. If we include the original demonstration, we now have results from error models whose ratios range from no coherent noise (maximum H error of 0 and maximum S error of .001) to purely coherent noise. \n\nAs shown in Fig. 3, as we increase the ratio of coherent to stochastic errors, the CNNs’ performances begin to diverge from the qpa-NNs' performances, before ultimately stabilizing at a statistically significant worse prediction accuracy. These results confirm our claim that the qpa-NNs outperform the CNNs, in part, because of their improved ability to model the effect of coherent errors on process fidelity.", "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": "", "extra_scores": {}}, {"review_id": "9iBx2B78Pc", "reviewer": "Reviewer_beTU", "summary": "This paper introduces a neural-network-based architecture for quantum capability learning. The idea is to utilize the architecture to predict rates of errors in quantum circuits. The authors compared their qpa-NN with previous CNN-based method and elucidated their improved performance.", "questions": "1. Have the authors compared their qpa-NN with the so-called “stability baseline model”, (which was mentioned to be better than CNN-based method Hothem et al. [2023c]) ?", "limitations": "There does not seem to be negative social impact of this theoretical research.", "rating": 5, "confidence": 3, "soundness": 3, "presentation": 3, "contribution": 2, "strengths": "1. qpa-NN outperforms the previous CNN method in predicting circuit success probability.", "weaknesses": "1. Given that neural network-based methods for predicting circuit success probability have been previously discussed in the literature, the novelty of the approach is not sufficiently convincing. \n2. The explanation of how the qpa-NNs leverage graph structures could benefit from further elaboration to enhance clarity. \n3. The scope of the considered noise types is limited, and the authors have benchmarked their results exclusively on small-scale devices.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "We are grateful for the reviewer's summary of our paper and their recognition of its strengths, especially our superior performance compared to CNNs. Your review and constructive feedback will be instrumental in improving our manuscript, significantly enhancing its clarity and impact.\nIn response to your feedback, we plan to include the following revisions in our final manuscript:\n   - A new large-scale demonstration of a qpa-NN trained to predict the process fidelity of 100-qubit circuits run on a simulated quantum computer experiencing low-levels of coherent and stochastic errors. This demonstration conclusively shows that the qpa-NNs scale to large-scale systems even when predicting process fidelity.\n   - An additional paragraph outlining the novelty of our work.\n   - A re-written Section 3.1 that better explains how our qpa-NNs leverage graph structures to reduce their parameter counts and make accurate predictions of a circuit’s fidelity.\n   - Results from three new 4-qubit demonstrations using three new error models.\n\nWe now address the four listed weaknesses and explain how our new results/edits address each of the weaknesses. We also include a comparison to the stability baseline model (SBM) from Hothem et. al. [2023c].\n\n**Weakness 1**: the lack of a large-scale demonstration (same as the response to reviewer k1xb).\n\nWe agree with the reviewer that the lack of a large-scale demonstration is the main weakness of our paper. We hope that our new 100-qubit demonstration will satisfy the reviewer. We believe that the qpa-NNs’ strong performance on this data conclusively demonstrates the scalability of our approach.\n\nFig. 1 in the rebuttal material depicts the results from training a qpa-NN to predict the process fidelity of circuits run on a noisy 100-qubit quantum computer experiencing low-levels of weight-1 coherent and stochastic errors. Because of the infeasibility of performing a strong simulation of coherent errors on a 100-qubit device, we used a first-order approximation to the strong simulation, like the technique used in the non-Markovian simulations in [Hothem et. al., 2023]. The qpa-NN achieves a MAE of .097%. This result shows that it is possible to construct qpa-NNs for large system sizes, and that they can achieve excellent prediction accuracy at large system sizes.\n\n**Weakness 2**: novelty.\n\nWhile we appreciate the reviewer’s concerns about our work’s novelty, we disagree with their assessment. In particular:\n   - Our approach is fundamentally different from past works in that it uses bespoke networks inspired by an in-depth understanding of the underlying physics of quantum computers. This innovation is akin to introducing physics-informed neural networks for solving PDEs or CNNs for image recognition, but for a more specific task.\n   - Moreover, we are the first to apply any kind of neural network to predicting process fidelity. Unlike PST, process fidelity is defined for any quantum gate, circuit, or channel and is the metric of choice for reporting gate and circuit performance. It is also estimated by many popular benchmarking protocols (e.g., randomized benchmarking). Our work thus addresses a widely applicable problem that is of general interest to the quantum computing community.\nWe will clarify the novelty of our work by adding a paragraph outlining our contributions in the next draft.\n**Weakness 3**: unclear explanation of how the qpa-NNs leverage graph structures.\n\nOur qpa-NNs leverage a graph structure to greatly reduce their size, enabling the construction of qpa-NNs for modelling many-qubit systems. Our qpa-NNs are scalable because they model a polynomial-sized set of error types, and they use a graph to represent which parts of a quantum circuit a particular error’s rate (at circuit layer\ni) will plausibly depend on (which specifies the input to each that error’s associated subnetwork). This error rate dependency is typically associated with the physical layout of a quantum computer’s qubits, and so we use the graph encoding of this layout to model this dependency. We plan to revise our description of how our qpa-NNs leverage graph structures to better explain this.\n\n**Weakness 4**: limited noise models.\n\nWe plan to include results from three new 4-qubit demonstrations to allay the reviewer’s concern that we only evaluated our qpa-NNs on a limited number of error models. Copying from our response to reviewer oCVW: \n\nWe performed three new 4-qubit demonstrations to better support our claim that the qpa-NNs improved performance is due to their  ability to model coherent noise. Each new demonstration used the same setup as the original 4-qubit demonstration, but with a different error model. The error models differed in the ratio of total coherent to stochastic noise allowed in each gate’s error model. If we include the original demonstration, we now have results from error models whose ratios range from no coherent noise (maximum H error of 0 and maximum S error of .001) to purely coherent noise. \n\nOur final manuscript will include a broad suite of realistic noise models, ranging from complicated experimental noise models to simple noise models for 100-qubit quantum computers.\n\n**Additional discussion**: comparison to the stability baseline model (SBM).\n\nThe qpa-NNs perform favorably compared to the SBM in Hothem et al. [2023c]. Fig. 2 in the rebuttal document shows that the qpa-NN outperforms the SBM on one device, achievers peer performance on three devices, and slightly worse performance on two devices.\n\nHowever, it is a category mistake to compare the SBM to the qpa-NNs because the SBM is not a predictive model of a device’s capability. The SBM is constructed by rerunning each circuit and comparing the results to those from the original run. Therefore, it quantifies how stable the device is by measuring how a circuit’s fidelity changes over time. Alternatively, it measures how well a device’s past performance predicts its future performance.", "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": "", "extra_scores": {}}, {"review_id": "Nc9cBPk18M", "reviewer": "Reviewer_k1xb", "summary": "The paper introduces a novel quantum-physics-aware neural network (qpa-NN) architecture for quantum capability learning. The model achieves error reduction in capability prediction on both experimental and simulated data.", "questions": "How does the proposed architecture handle scalability challenges as the number of qubits increases?", "limitations": "The authors have discussed the limitations.", "rating": 6, "confidence": 3, "soundness": 3, "presentation": 3, "contribution": 2, "strengths": "1. The qpa-NN architecture incorporates quantum physics principles, which provides a new perspective of designing models in the field of learning-based quantum capability learning.\n\n2. The demonstrated reduction in mean absolute error over CNN-based models is commendable.", "weaknesses": "1. The reviewer's main concern is regarding the qubit scale of the dataset. However, the dataset used in the experiments, which consists of maximum 5 qubits, appears too small both from the perspective of current quantum hardware and classical simulations. The reviewers are curious to know the reason behind the inability to collect data on systems with more qubits (either experimental data or simulated data). Is it due to the high difficulty of experimental deployment, or is it because the current models struggle to train on larger datasets?\n\n2. The evaluation method mentioned in the paper, Process Fidelity (Eq. 3), may have scalability issues. Although the authors attempted to explain in Section 3.2 how some approximations can be used to calculate Eq. 3 relatively efficiently, this description is hard to follow. For instance, can the proposed approximation method avoid exponential computational and storage complexity? Additionally, the reviewer hope that in the revised version, the authors can provide a discussion on the differences between the approximations designed in this paper and the estimation methods in [Proctor et al., 2022], or if the approximations in this paper are merely direct adaptations of the latter's estimation methods.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "We appreciate the reviewer’s summary of our paper and their recognition of its strengths. Your thorough review and constructive feedback have been invaluable in refining our manuscript. Incorporating your suggestions will greatly enhance its quality and impact.\n\nIn response to your feedback, we plan to include the following revisions in our final manuscript:\n   - A new large-scale demonstration in which we train a qpa-NN to predict the process fidelity of 100-qubit circuits executed on a simulated quantum computer experiencing low-levels of coherent and stochastic errors. We believe that this demonstration conclusively shows that the qpa-NNs scale to large-scale systems even when predicting process fidelity.\n   - A short discussion on how our qpa-NN approach to capability learning differs from, yet is synergistic with, the mirror circuit fidelity estimation (MCFE) protocol introduced in [Proctor et. al., 2022].\n   - Clearer exposition on how our qpa-NNs estimate process fidelity without exponential and computational storage complexity by only predicting the most influential terms in the first-order approximation to process fidelity (Eq. 3).\n\nWe now address, in order, the three listed weaknesses and explain how our new results/edits address each of the weaknesses.\n\n**Weakness 1**: the lack of a large-scale demonstration (same as the response to reviewer beTU).\n\nWe agree with the reviewer that the lack of a large-scale demonstration is the primary weakness of our paper, and we hope that our new 100-qubit demonstration will satisfy the reviewer. We believe that the qpa-NNs’ strong performance on this data conclusively demonstrates the scalability of our approach.\n\nFig. 1 in the rebuttal material depicts the results from training a qpa-NN to predict the process fidelity of circuits run on a noisy 100-qubit quantum computer experiencing low-levels of weight-1 coherent and stochastic errors. Because of the infeasibility of performing a strong simulation of coherent errors on a 100-qubit device (fully modelling coherent errors is equivalent to universal quantum computation), we used a first-order approximation to the strong simulation, like the simulation technique used in the non-Markovian simulations in [Hothem et. al., 2023]. The qpa-NN achieves a mean absolute error of .097%. Crucially, these results demonstrate that is both possible to construct qpa-NNs for arbitrarily large system sizes, and that the qpa-NNs can achieve excellent prediction accuracy at arbitrarily large system sizes.\n\n**Weakness 2**: unclear explanation of scalability.\n\nIn addition to including two new empirical demonstrations of our the qpa-NNs’ scalability, we plan to clarify the theoretical arguments for the scalability of qpa-NN’s in the text. There are two primary criteria for developing a scalable machine learning technique to assess the performance of a quantum computer. The first is the ability to efficiently (i.e., without exponential time or sample complexity) gather training data and the second is ensuring that the model’s size grows polynomially in the device size. The qpa-NN approach satisfies both criteria.\n\nWe can efficiently gather training data for qpa-NNs trained on experimental hardware, regardless of if we are using probability of successful trials (PST) or process fidelity. A definite-outcome circuit’s PST can be efficiently estimated to 1/sqrt(N) precision by running the circuit N times on the quantum computer, while any circuit’s process fidelity can be estimated to O(1/sqrt(N)) precision by running three ensembles of closely related circuits O(N) times using MCFE [Proctor et. al., 2022]. The only limiting factor to gathering training data on real hardware are the noise levels in current real-world, large-scale devices. \n\nLikewise, we limit the growth of our qpa-NNs’ parameter counts to be polynomial in the number of qubits by only tracking the effect of the most important errors in a device, the local, low-weight errors. Physics intuition tells us that if a gate G acts on qubit Q, then the most probable errors induced by G will affect a local neighborhood of Q. Moreover, these errors are unlikely to affect too many qubits at once. In other words, gates induce low-weight, local errors, as determined by a device’s connectivity graph. Our qpa-NNs reflect this intuition by only trying to predict the impact of local, low-weight errors on a circuit’s fidelity and completely ignoring the contributions of the highly improbable high-weight and non-local errors. \n\nAs a result, a qpa-NN’s parameter count grows polynomially in device size. See Fig. 4 for a visualization of how a qpa-NN’s parameter count grows with device size for a fixed set of hyperparameters. For a device with a grid connectivity graph, the parameter count grows quadratically, while it grows linearly for a device with a ring connectivity graph.\n\n**Weakness 3**: lack of comparison with mirror circuit fidelity estimation (MCFE).\n\nWe thank the reviewer for highlighting this point of confusion. It appears that we did not clearly explain how our method differs from the method of mirror circuit fidelity estimation (MCFE) proposed in [Proctor et. al., 2022]. We will clarify how our method for training qpa-NNs to predict process fidelity scales and differs from MCFE.\nMCFE and qpa-NNs solve different problems. qpa-NNs aim to construct predictive models of a quantum computer’s capability, predicting the process fidelity of any new circuit from a class after training on a representative sample. MCFE, however, estimates the process fidelity of a known target circuit by running related circuits on a quantum computer. While MCFE does not provide a model, it is useful for gathering training data to build predictive models.", "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": "", "extra_scores": {}}]

Author rebuttal:
We sincerely thank the referees for their time and insightful comments on our manuscript. Your thorough review and constructive feedback have been invaluable in identifying areas for improvement and clarity. We are confident that incorporating your suggestions will significantly strengthen the quality and impact of our paper, making it worthy of inclusion at NeurIPS 2024.

In response to each reviewer’s feedback, we plan to modify our paper to:
   - Better support our claim that our qpa-NN approach scales by including a new large-scale demonstration on a simulated 100-qubit computer.
   - Better support our claim that the qpa-NNs’ improved performance is due to their improved modeling of coherent errors by adding an appendix with results from three new simulated 4-qubit devices, each experiencing different ratios of coherent to stochastic error strengths.
   - Clarify how our approach differs from, yet synergizes with, mirror circuit fidelity estimation (MCFE) [Proctor et. al., 2022] by adding a few sentences to Section 2.2.
   - More clearly explain how our approach avoids the exponential scaling afflicting other methods for predicting process fidelity by rewriting Section 3.2.
   - Clarify how our qpa-NNs exploit the graph structure of a quantum computer’s native connectivity graph to efficiently track and model the effects of only the most relevant errors by rewriting Section 3.1.
   - Better explain the novelty of our results by adding a paragraph to the introduction clearly stating this work’s novel contributions to the literature.

Additionally, we have provided a comparison between the qpa-NNs and the stability baseline model (SBM) in [Hothem et. al., 2023c] in our rebuttal. We chose not to include these results in our revised paper as the SBM is not a predictive model and its inclusion would muddle the paper’s presentation. Nonetheless, we strongly believe that our revisions comprehensively address each reviewer’s critiques and that our final product will merit inclusion in this year’s NeurIPS. Lastly, we include an attached pdf with supporting figures.

We now briefly outline how our proposed revisions address specific reviewer critiques. We provide full discussions in our responses to each individual reviewer.
   - Our 100-qubit demonstration should allay reviewer beTU’s and k1xb’s concerns about analyzing only small-scale devices. As shown in Fig. 1 of the rebuttal PDF, our trained qpa-NN obtained a mean absolute error of .097% when predicting the process fidelity of 100-qubit circuits run on a simulated 100-qubit quantum computer experiencing weight-1 coherent and stochastic errors. Further details are provided in our rebuttals to each reviewer. Unfortunately, it is not possible to perform a 100-qubit demonstration on real hardware as IBM’s cloud-accessed, 127-qubit processors are currently too noisy to reliably execute circuits with high fidelity.
   - New 4-qubit simulations should satisfy reviewer oCVW’s desire for a more thorough investigation of whether the qpa-NNs’ improved performance is due to their improved ability to model coherent errors, as well as reviewer beTU’s concern about the scope of the considered noise types. In our new 4-qubit demonstrations, we repeated the 4-qubit demonstration in the paper using three new error models, each with a different ratio of coherent to stochastic errors. As shown in Fig. 3, as we increase the ratio of coherent to stochastic errors, the CNNs’ performances rapidly deteriorate, while the qpa-NNs’ performances ultimately stabilize. These results confirm our claim that the qpa-NNs outperform the CNNs, in part, because of their improved ability to model the effect of coherent errors on process fidelity.
   - Revisions clarifying how our approach differs from, yet synergizes with, MCFE address reviewer k1xb’s request for clarification. Our revisions more clearly explain the different goals of our qpa-NN approach—to build predictive models of a quantum computer’s capability that can predict a circuit’s fidelity without running the circuit—and MCFE—to estimate the fidelity of select circuits by running those circuits on the quantum computer. Our revisions also clarify how to use MCFE to efficiently gather training data for the qpa-NNs, a necessary step in a scalable pipeline for building predictive models.
   - Revisions clarifying our approach to predicting process fidelity address reviewer k1xb’s concern that our approach may have scalability issues. In our revisions, we explain how we avoid the exponentially expensive cost of evaluating equation 3 by focusing solely on predicting the effect of the most important errors in a quantum computer—local, low-weight errors—on process fidelity. Because the number of local, low-weight errors scales polynomially with device size, our qpa-NNs’ parameter counts also scale polynomially with device size. Put another way, we design qpa-NNs to approximate eq. 3 by only predicting a polynomial-sized set of the most important terms in equation 3.
   - Revisions clarifying how our qpa-NNs use a graph structure to encode the spatial dependencies of a quantum computer’s error structure, and how this enables our qpa-NNs to have a small, polynomial number of parameters. 
   - Revisions more clearly stating our contributions address reviewer beTU’s concern that our approach is not “sufficiently novel.” As explained in our rebuttal, we introduce a novel, sophisticated neural network architecture for solving the quantum capability learning problem and are the first group to tackle the problem of predicting process fidelity, the most widely used circuit-success metric in the quantum computing community.

Meta-review: The paper introduces a neural network to predict quantum capability, namely quantifying the noise in the quantum circuit. The work focuses on Clifford gates which can be simulated efficiently and presents experiments on quantum devices as well. The method improves the state of the art for predicting the behaviour of a quantum computer using classical neural networks, an important direction in quantum computing. The rebuttal addresses some of the questions on scalability and related work.
Decision: Accept (poster)