Variational quantum eigensolvers (VQEs) are leading candidates to demonstrate near-term quantum advantage. Here, we conduct density-matrix simulations of leading gate-based VQEs for a range of molecules. We numerically quantify their level of tolerable depolarizing gate-errors. We find that: (i) The best-performing VQEs require gate-error probabilities between 10^{−6} and 10^{−4} (10^{−4} and 10^{−2} with error mitigation) to predict, within chemical accuracy, ground-state energies of small molecules with 4 − 14 orbitals. (ii) ADAPT-VQEs that construct ansatz circuits iteratively outperform fixed-circuit VQEs. (iii) ADAPT-VQEs perform better with circuits constructed from gate-efficient rather than physically-motivated elements. (iv) The maximally-allowed gate-error probability, _{c}, for any VQE to achieve chemical accuracy decreases with the number _{II} of noisy two-qubit gates as _{c} decreases with system size, even with error mitigation, implying that larger molecules require even lower gate-errors. Thus, quantum advantage via gate-based VQEs is unlikely unless gate-error probabilities are decreased by orders of magnitude.

Calculating the ground-state energy of a molecular Hamiltonian is an important but hard task in computational chemistry^{1}. For strongly correlated systems, exact classical approaches quickly become infeasible as system sizes exceed 100 spin-orbitals. Other, approximate, methods often lack accuracy^{1–5}. This makes quantum computers an attractive alternative. A potential route to quantum-chemistry simulations relies on the quantum-phase-estimation algorithm (QPEA)^{1,6}. However, the QPEA requires executing millions of gates on error-corrected hardware^{7}. Realizing such hardware requires significant resource overheads and gate-error probabilities below a minimal threshold^{8}. For example, the surface code^{9,10}, requires thousands of physical qubits to implement a single logical qubit at a gate-error probability of 10^{−4} ^{11}. In view of these requirements, the QPEA is not yet feasible.

To reduce the qubit number and gate-error requirements, the variational quantum eigensolver (VQE) was proposed^{12}. The VQE is a hybrid quantum-classical algorithm that uses a classical optimizer and a parameterized quantum circuit, the ‘ansatz’, to estimate ground-state energies. Combined with significant hardware developments^{13}, VQEs have facilitated successful demonstrations of quantum computational chemistry for small systems^{12,14–18}. These demonstrations have been aided by VQE algorithms’ abilities to correct for certain errors^{15,19,20}. Despite these achievements, there are still significant hurdles to overcome for VQEs to become useful. First, the short, gate-efficient ansätze used in small-scale experimental demonstrations^{12,14–18} face optimization difficulties for larger systems. This is related to the emergence of barren plateaus (vanishing gradients), which are more likely when the ansatz is unrelated to the Hamiltonian^{21,22}. Current research, for example on growing the ansatz circuit ^{23}) is aimed at avoiding or mitigating the issue of barren plateaus^{21}. Another significant hurdle comes from gate-error rates in hardware. Although current noisy intermediate-scale quantum devices^{24–26} have sufficiently many qubits to run VQEs for molecules with more than 100 spin-orbitals^{13}, their gate-error rates are too high.

At present, efforts to ameliorate the gate-error issue aim to either reduce ansatz circuit depths^{23,27–30} or implement elaborate error-mitigation schemes^{31–33}. However, VQEs are often benchmarked in the absence of gate errors, with circuit depths and CNOT counts used as proxies of their noise resilience^{30}. It has been argued that the maximum viable circuit depth for a VQE ansatz circuit is given by the reciprocal of gate-error probability 1/^{1}. More rigorously, given a gate-error probability ^{34,35}. A research question, which remains under-explored, is to quantify the gate-error probabilities that VQEs can tolerate. Specifically, considering the analogy of a surface code, which has a well-defined fault-tolerance threshold^{11}, we aim to find the maximally allowed gate-error probability, below which a certain VQE estimates a certain molecule’s energies within chemical accuracy. Quantifying the maximally allowed gate-error probability allows the noise resilience of leading VQEs to be ranked, and provides useful goals for the hardware community.

In this article, we numerically quantify under how high gate-error probabilities VQEs can operate successfully. More specifically, using density-matrix simulations, we simulate the ground-state search of leading, gate-based VQEs for a range of molecules. In the presence of depolarizing noise, we show that: (i) Even the best performing VQEs require gate error probabilities _{c} on the order of 10^{−6} to 10^{−4} (without error mitigation) in order to predict molecular ground-state energies within chemical accuracy of 1.6 × 10^{−3} Hartree. This is significantly below the fault-tolerance threshold of the surface code^{11}. For small systems, error mitigation can be employed such that the required _{c} values can be improved to 10^{−4} to 10^{−2}. (ii) ADAPT-VQEs tend to tolerate higher gate-error probabilities than VQEs that use fixed ansätze, such as UCCSD and k-UpCCGSD. (iii) ADAPT-VQEs tolerate higher gate-error probabilities when circuits are synthesized from gate-efficient^{27–29,36}, rather than physically-motivated^{23}, elements. We support these claims by estimating, in the presence of depolarizing noise, the scaling relation between the maximally tolerable gate-error probability _{c} and the number _{II} of noisy (two-qubit) gates. Our results indicate that _{c}, decreases with system size, with and without error mitigation. This shows that larger molecules would likely require even lower gate-errors. We conclude that substantial quantum advantage in VQE-based quantum chemistry is unlikely, unless gate-errors are significantly reduced, or error-corrected hardware is realized, or error-mitigation protocols are improved and made scalable.

In this work, we investigate several classes of VQE algorithms. Our study prioritizes VQEs with short ansatz circuits, as these are expected to be more noise resilient^{29,30}. Specifically, we consider ADAPT-VQEs, which have comparatively short ansatz circuits^{23,29} and the ability to mitigate rough parameter landscapes^{21}. We further consider UCCSD^{37} and k-UpCCGSD^{38} as prototypes of fixed ansatz VQEs – the latter for its comparatively shallow ansatz circuits^{30}. Before we outline the results of our noise-resilience investigation, we describe the workings of the (ADAPT-)VQE.

The main idea of VQEs is to use shallow ansatz circuits, defined by a set of parameters ^{1}. ADAPT-VQEs use a classical optimizer in two ways^{23}: to conduct the Rayleigh-Ritz minimization with respect to a parameterized quantum state; and to iteratively construct the ansatz that generates the parameterized state itself. A quantum computer is used to calculate the energy-expectation value of the parameterized state.

Consider the state generated by the ansatz _{n}:_{n}(_{1}, …, _{n}) such that _{1} > … > _{n} and _{n} approaches

The iterative ansatz construction proceeds as follows. First, the ADAPT-VQE algorithm initializes a state _{0}, usually the Hartree-Fock state^{2}. Then, the algorithm generates a sequence of trial states by successively adding elements of the form_{α}, for _{α}(_{n}) in the

After choosing the _{n}(_{n}), a classical computer optimizes and updates the parameters _{1}, …, _{n} to minimize the energy expectation value _{n} in Eq. (_{n} − _{n−1} > _{n} − _{n−1} ≤ _{N} ≡ _{n} as the estimate of

In this work, we focus on the three main types of ADAPT-VQEs: fermionic-ADAPT-VQE, QEB-ADAPT-VQE and qubit-ADAPT-VQE. (Efficient gate-representations for their relevant ansatz elements can be found in refs. ^{28,29}). These algorithms differ in their ansatz-element pools

First, we consider the fermionic-ADAPT-VQE^{23}. As the name suggests, this algorithm uses a pool of operators that closely simulate the physics of fermionic excitations. The pool is formed from_{i} are fermionic creation and annihilation operators acting on the ^{39}, where_{i}, _{i}, _{i} are the Pauli operators acting on the ^{23}. Further, choosing fermionic excitations along the gradient of minimum energy produces ^{21}.

Second, we consider the QEB-ADAPT-VQE^{29,36}. This algorithm uses a pool of operators that nearly (up to a ± sign) simulate the physics of fermionic excitations. The pool is formed from_{i} are known as qubit creation and annihilation operators, respectively. Due to the CNOT efficiency of its pool, the QEB-ADAPT-VQE can find ground-state and excited-state energies with fewer CNOT gates than the fermionic-ADAPT-VQE^{28,29,36}.

Finally, we consider the qubit-ADAPT-VQE^{27}. This algorithm uses a pool of gate-efficient elements without physical motivation. The pool is formed from segments of Pauli-operator strings:_{i} denotes Pauli operators _{i}, _{i}, _{i} acting on the ^{27}. In our simulations, we use a pool formed from ^{27}, but at the expense of reduced circuit efficiency of the final ansatz^{29}.

Typically, the fermionic-ADAPT-VQE and the qubit-ADAPT-VQE use the gradient-based decision rule expressed in Eq. 8. On the other hand, the original QEB-ADAPT-VQE uses the energy-based decision rule, shown in Eq. 9. These algorithms are summarized in a flow-chart summary in Fig.

At each iteration, an ansatz element is chosen according to one of the two decision rules defined in green below the chart. This element is appended to the ansatz, the parameters are optimized, and the energy expectation value is estimated. The algorithm halts when the change in energy between iterations is below a given threshold.

To demonstrate the benefits of iteratively-grown ansätze, we compare them to a typical fixed-ansatz-VQE method: the UCCSD-VQE^{17,37}. In Supplementary Note ^{38}. Owing to its linear scaling of circuit depth with qubit number, this algorithm was recently put forward as the leading fixed-ansatz VQE^{30}. We simulate the workings of the fixed-ansatz methods using the aforementioned fermionic and QEB elements.

Given the breadth of work on VQEs^{30}, it is not possible to perform an exhaustive analysis of all existing algorithms. Nevertheless, the analytical results in Sec. II E, and the low circuit depths provided by ADAPT-VQE^{30}, suggest that our results provide a lower bound on the requirements for gate-based VQE algorithms to operate successfully. However, there exist algorithms that differ greatly from typical VQEs, and could deserve future attention. We discuss some of these, and the reasons for our exclusion of them, below. We will not consider iterative qubit coupled cluster (iQCC)^{40} and ClusterVQE^{41} algorithms. We do not anticipate these algorithms to be feasible options to study strongly correlated systems, whose simulation using quantum algorithms provides the most benefit over classical algorithms. We also omit the DISCO-VQE^{42}. Due to its large jumps in Hilbert space during the discrete optimizations of the ansatz, we expect DISCO-VQE to lack tolerance to barren plateaus. These problems may be overcome in future improvements of these VQE algorithms. We leave the design of improved algorithms, and the noise-evaluation of them, to future articles. Finally, we omit the ctrl-VQE algorithm^{43}. Although highly interesting, this Hamiltonian algorithm operates with device-tailored pulses, rather than quantum gates, and thus lies outside the scope of this work.

To investigate the effect of noise on gate-based VQE, we constructed a VQE-tailored density-matrix simulator, expanding the state-vector circuit simulator of ref. ^{29}. We represent molecular orbitals in the Slater type orbital-3 Gaussians (STO-3G) spin-orbital basis set^{44}, with the option of frozen orbitals. The openfermion-Psi4 package^{45,46} is used to generate the second-quantized Hamiltonian and to perform the Jordan-Wigner transformation^{39}. Ansatz parameters are optimized using Nelder-Mead^{47} or gradient-descent-based (BFGS)^{48} methods in SciPy^{49}.

We note that, due to the wide array of quantum-computing platforms and their contrasting qubit-control implementations, no noise model can be simultaneously realistic and platform-agnostic. In this work, we model noise by applying single-qubit depolarizing noise to the target qubit

In real devices, noise from two-qubit gates completely dominates the noise from single-qubit gates^{25,50–52}. Thus, we ignore the latter. Additionally, we exclude state preparation and measurement errors, which are often lower in magnitude than the accumulated two-qubit gate errors^{50}, and can be mitigated efficiently in experiments^{53–58}. (We note that ADAPT-VQE algorithms have high measurement requirements, such that measurement errors may prevent the algorithm from reaching the global minimum energy. This topic requires further investigation). Depolarizing noise is commonly used to represent local and Markovian gate errors when assessing both NISQ^{59–61} and quantum-error-correction^{11,62–64} algorithms. More realistic models can include thermal-relaxation noise (dephasing and amplitude damping)^{65} and device-specific gate errors derived from gate-set-tomography data^{66}. When _{1} ≈ _{2} thermal relaxation noise can be approximated using our depolarizing noise model^{62}. This is a reasonable model for superconducting hardware^{50}. On the other hand, when _{2} ≪ _{1} dephasing noise dominates and our depolarizing noise model is less accurate. This is common in trapped-ion devices^{51,52} and spin qubits^{67} and has recently been investigated in ref. ^{68}. Moreover, existing VQE algorithms require unrealistically low error rates to give chemically accurate energies. Any attempt to scale down the error rates in realistic noise models to these low levels must be theoretically-justified. This is challenging for a complex, multi-parameter model. Hence, we exclude noise models based on gate-set tomography. Finally, we do not consider coherent errors, since their effect can be suppressed by randomized compiling^{69} and dynamical decoupling^{70,71}. Randomized compiling^{72} can also be used to convert coherent errors to stochastic errors. Additionally, VQE algorithms are somewhat resilient to coherent errors^{12,73}. Thus, in this work we focus on incoherent errors. Note that, if VQE algorithms were studied with a coherent noise model, their perceived performance may be greater.

When simulating the smallest molecules (H_{2} and H_{4}) we apply our noise channel [Eq. (_{4}, we approximate each noisy ansatz element by a corresponding noiseless ansatz evolution and a noise-inducing evolution. The noise-inducing evolution corresponds to depolarizing noise applied to each qubit in accordance with the number of times that qubit was a CNOT target in the ansatz element. We observe that this lower-bounds the effect of noise. For example, for H_{4}, applying noise after each CNOT with gate-error probability _{2} and H_{4}. A detailed illustration of our noise approximation is given in Supplementary Note

Energy accuracy is the key metric of VQE performance. It is defined as_{n}(_{FCI} is the energy given by the full-configuration-interaction^{5} calculation of the true ground-state energy _{c} for which Δ

Classical optimizers are used to tune _{1}, …, _{n}. The parameters are optimized until the gradient norm, _{O}. In our simulations of H_{2} we calculate converged values of _{n}(_{t}: Δ_{t}. Then, we approximate Δ

In this section, we benchmark the noise resilience of ADAPT-VQEs using our density-matrix simulator. We study H_{2}, H_{4}, LiH, HF and BeH_{2}. Our simulations were conducted using a parameter optimization cut-off of _{n} − _{n−1} ≤ ^{−12} Hartree. In our simulations of the larger molecules, we used an ansatz-truncation cut-off of Δ_{t} = 10^{−4} Hartree. Below, we use Δ_{final}) to refer to the energy accuracy at the final ansatz length _{final}. Because of the significant skepticism towards error-mitigation strategies^{35,74,75}, we omit such strategies from the analyses presented in this section and investigate error mitigation separately in Sec. II F.

The inset of Fig. _{2}. The values of ^{8,11} as well as the gate-error probability of currently available quantum hardware^{25,50–52}. All tested VQE algorithms require extremely small gate-error probabilities if they are to improve on the Hartree-Fock energy approximation, even for the simple H_{2} molecule. The region of chemical accuracy is too small to show in the inset. In real implementations of ADAPT-VQE algorithms, energies exceeding the Hartree-Fock approximation would not be achieved, since, in this case, adding elements to the ansatz does not improve the initial energy accuracy. Here, we add noise to a noiselessly-grown ansatz such that these energies are shown. These observations motivated us to reduce significantly the range of

Plotted for H_{2} (_{4} (at 1 Å (_{2} (^{17,37}. Energy accuracies lower than chemical accuracy are highlighted by the yellow region. The purple line in the H_{2} inset is the energy calculated using the Hartree-Fock^{2} state. Extrapolated noise-susceptibility calculations are shown in black for the fermionic-ADAPT-VQE (

The rest of Fig. _{c} for computing ground-state energies within chemical accuracy are extremely small. For all molecules investigated in this study, the value of _{c} is on the order of 10^{−6} to 10^{−4} (see Table _{2} and H_{4} (1 Å) suggest that ADAPT-VQEs outperform fixed ansatz methods. For a given pool of ansatz elements, the corresponding ADAPT-VQE algorithm leads to better energy accuracies than the corresponding fixed ansatz VQE algorithm. Third, the efficient representation of fermionic excitations^{28} improves the performance of the fermionic-ADAPT-VQE significantly. This representation reduces CNOT depth, but its scaling of CNOT depth with molecule size is still worse than the scaling of QEB and Pauli string elements. The second and third observations support the claim^{29} that the CNOT count is a useful estimator of VQE’s noise vulnerability. Fourth, the more gate-efficient (Pauli string and QEB) pools outperform the most physically-motivated (fermionic) pool. The fermionic-ADAPT-VQE is consistently outperformed by either the qubit-ADAPT-VQE or the QEB-ADAPT-VQE. Fifth, sometimes the QEB-ADAPT-VQE outperforms the qubit-ADAPT-VQE and vice versa. For H_{2}, H_{4} (1 Å) and BeH_{2}, the qubit-ADAPT-VQE outperforms the QEB-ADAPT-VQE. On the other hand, for HF, the QEB-ADAPT-VQE (energy-based decision rule) outperforms the qubit-ADAPT-VQE. For LiH, the QEB-ADAPT-VQE and the qubit-ADAPT-VQE perform similarly. Notably, for H_{4} (3 Å), the qubit-ADAPT-VQE fails to add more than two elements to the ansatz. Hence, it never surpasses chemical accuracy. This gives some indication that the qubit-ADAPT-VQE is worse than the QEB-APAPT-VQE at simulating strongly correlated molecules. Sixth, different decision rules for QEB-ADAPT-VQEs yield different performances. For HF, LiH and BeH_{2}, the energy-reduction decision rule gives a better energy accuracy than the maximum-gradient rule. Conversely, for H_{4} (1 Å and 3 Å) the gradient-based decision rule performs better. A study of the optimal decision rules for various molecular-energy landscapes is left for the future.

Maximum gate-error probabilities _{c} [ × 10^{−5}] for which chemical accuracy is achieved.

Molecule (Separation) | H | H | H | LiH | HF | BeH | H | H | LiH | H | |
---|---|---|---|---|---|---|---|---|---|---|---|

Fermionic-ADAPT-VQE | 5.30 | 0.30 | 0.36 | 0.52 | 0.19 | 0.11 | 0.30 | 0.39 | 0.96 | 0.04 | 0.06 |

Efficient fermionic-ADAPT-VQE | 5.30 | 1.38 | 1.34 | 2.02 | 0.89 | 0.49 | 1.51 | 1.52 | 3.47 | ||

QEB-ADAPT-VQE (Energy) | 23.59 | 1.58 | 1.05 | 4.14 | 4.15 | 0.68 | 1.72 | 2.53 | 7.69 | 0.35 | |

QEB-ADAPT-VQE (Gradient) | 23.59 | 1.73 | 1.45 | 2.25 | 0.97 | 0.59 | 1.73 | 1.64 | 3.92 | 0.32 | |

qubit-ADAPT-VQE | 41.20 | 2.19 | N/A | 4.28 | 2.15 | 0.75 | 2.23 | N/A | 8.66 | 0.30 |

The first six columns present data from Fig.

We close this subsection with a comment on benchmarks of fixed-ansatz VQEs. Both UCCSD and k-UpCCGSD ansätze have been investigated with numerical simulations. However, their energy accuracies significantly worse than those obtained using ADAPT-VQEs. In particular, the energy accuracies for the k-UpCCGSD algorithm do not fit the scale of Fig.

In the above analyses of ADAPT-VQEs for larger molecules, we truncated noiselessly-grown ansätze in the _{t} was reached. Thus, we established a performance hierarchy between different VQEs. However, deeper circuits are generally more vulnerable to noise, and fixing _{4} (1 Å), H_{4} (3 Å) and LiH in the left, middle and right columns of Fig. _{opt}) as a function of

Plotted for three different molecules: H_{4} (at 1 Å (

The data in Fig. _{4} (3 Å), which simply fails after _{opt}(_{opt}(_{opt}(_{opt}(_{opt}) as a function of _{c} increase when the ansätze are truncated at an optimal value of _{c} are presented in Table ^{−6} to 10^{−4} Hartree.

To analytically support our numerical results, we study the linear response of energy accuracy Δ_{c} is roughly inversely proportional to the number _{II} of noisy (two-qubit) gates.

From Fig. _{II} (details are given in Supplementary Note _{R} ⋯ _{1}. We use _{r} is a noisy (CNOT) gate, and we use _{r} to denote the qubit which noise acts on. Further, we define a perturbed version of the target unitary _{r} after the _{U} is close to

The energy expectation values underlying _{c} for molecules too large to study with our density-matrix simulator. The estimates of _{c} for H_{2}0 and _{c}] which allow for chemically-accurate simulations.

The energy fluctuations are bounded by the spectral range of _{II}, as _{C} = 1.6 × 10^{−3} Hartree (chemical accuracy). This result is supported by recent results in condensed matter systems^{76}. The inverse proportionality between _{c} and _{II} suggests that gate-error probabilities will have to reach extremely small values for useful chemistry calculations with VQE algorithms to be viable. Alternatively, we require improved VQE algorithms with shallower circuits and fewer noisy (two-qubit) gates.

Noise susceptibility (top) and average energy fluctuation (bottom) as functions of the number _{II} of CNOT gates for all molecules and algorithms reported in Sec. II C, at all circuit depths.

In the absence of error-corrected hardware, several strategies to mitigate the effect of noise have been suggested^{31–33,77}. Quantum error mitigation is a family of strategies which generally rely on knowledge of a circuit, noise model, or both to generate a set of modified circuits. Sampling from these circuits can generate a better estimate of the noiseless circuit’s output^{77}. While these strategies have been demonstrated in simple VQE implementations^{14,78,79}, they suffer, in general, from exponential scaling of sample requirements with qubit number^{35,74,75}, potentially preventing their viability in useful NISQ VQE implementations. Indeed, leading reviews on quantum computational chemistry^{1}, state that ‘it seems unlikely that error-mitigation methods alone would enable more than a small multiplicative increase in the circuit depth.’ This unfavorable scaling has also been observed experimentally, where it has prevented the use of all but the most simple mitigation strategies^{26}.

The main goal of this work is to assess the required error rates for useful VQE implementations of molecules with more than 100 spin-orbitals. Due to the uncertainty around their scalability, as well as the unclear performance in the presence of time-dependent noise (particularly two-level-system defects^{80} which drift in frequency), a study of this type should not include quantum error mitigation in its current form. Despite this, we believe it is relevant to extend our study to ascertain the maximally allowed gate-error probability _{c} to calculate molecular energies within chemical accuracy for an error-mitigation protocol with polynomial sampling overhead. To partially address this question, we repeat our numerical simulations using linear zero-noise extrapolation^{31,32} with a noise multiplication factor of 3. Despite being biased and heuristic^{81}, we choose linear zero-noise extrapolation for its modest sampling overhead and numerical stability, which proved useful in recent large-scale demonstrations of error mitigation^{26}.

The results for H_{2}, H_{4} and LiH are depicted in Fig. _{c}-ranking does not change. This suggests that a VQE algorithm with higher noise resilience in the absence of error mitigation would remain more noise resilient when error mitigation is applied. Finally, we put the improved gate-error probabilities _{c} into context by plotting them as crosses in Fig. _{c} with the problem size _{c} sufficiently for useful system sizes,

Plotted for H_{2} (_{4} (at 1 Å) (_{2} (^{31}^{,}^{32}. Energy accuracies lower than chemical accuracy are highlighted by the yellow region. The crosses represent the intercepts of the curves with this region.

For molecules with the same number of orbitals, the mean probability is taken. The crosses and circles represent the noise probabilities required to reach chemical accuracy with and without error mitigation, respectively. The data without error mitigation is taken from Table ^{93}.

Any quantum algorithm aimed at near-term NISQ devices must be designed to tolerate some level of noise. In this work, we numerically quantify the maximally allowed depolarizing gate-error probabilities, _{c}, required by leading gate-based VQEs to achieve chemically accurate energy estimates. Based on numerical simulations, we reach five conclusions. First, even the best-performing VQE algorithms require gate-error probabilities between 10^{−6} and 10^{−4}, for the small molecules we assess. Such errors are at least an order of magnitude below state-of-the-art experiments^{25,50} and the surface-code threshold^{8,11}. If error mitigation is viable, the _{c} values can be improved to 10^{−4} to 10^{−2} with linear zero-noise extrapolation. Second, larger molecules tend to require longer ansatz-circuits and thus, lower gate-error probabilities, see Fig. _{c} than equivalent fixed-ansatz VQEs, including those with the shortest ansatz circuits^{30,38}. Fourth, the more gate-efficient the ADAPT-VQE ansatz pool, the more noise resilient the algorithm. From a noise-resilience perspective, qubit excitations and Pauli-string excitations outperform fermionic excitations. Fifth, the maximum gate-error probability allowed to reach chemical accuracy is roughly inversely proportional to the number of CNOT gates:

In this work, we quantify the maximally allowed gate-error probability of ADAPT-VQEs, UCCSD VQE and the leading fixed-ansatz VQE, k-UpCCGSD^{38}. The latter is chosen as due to its favorable circuit depth scaling with molecule size^{30}. This ignores plenty of other VQE algorithms which would benefit from similar studies in the future, as discussed in the main text^{40–43,82–87}.

As opposed to a fault-tolerance threshold in error correction, the maximally allowed gate-error probability _{c} crucially depends on the size of the input problem, see Fig. _{c} tends to shrink as the number of spin orbitals _{c} decreases with _{II} = poly(_{c} with the number of spin orbitals

While this study is entirely focused on gate-errors, other sources of noise may also be relevant. These include errors from state preparation and measurement as well as statistical noise due to sampling of expectation values from a limited number of shots. As mentioned when justifying the noise model, errors due to state preparation and measurement tend to be smaller than the accumulated gate errors, and there are widely-implemented methods to compensate for them^{53–58}. However, in principle, measurement errors may lead to sub-optimal parameter values or operator choices during ansatz growth of ADAPT-VQE, which may prevent the algorithm from reaching the global minimum energy. A detailed analysis of such effects is left for future work.

While it is possible to sample any expectation value with ^{1,30}, the scaling prefactor may lead to prohibitively large run-times^{30,88–90}. This issue is particularly acute for ADAPT-VQE algorithms, where each growth step requires shots for both parameter optimization and element selection. In this case, the number of necessary gradient measurements for each ansatz growth step is greater than that for VQE parameter optimization, by a factor which scales linearly in the number of qubits^{91}. Holistic studies of VQE run-times^{30,88–90} provide predictions which vary greatly depending on the estimation methodology. The estimated run-times are often intractable without significant parallelization. The number of necessary measurements for parameter optimization can potentially be reduced via alternate groupings of Pauli operators^{83–85}, or tensor contraction of the Hamiltonian (such as by double factorization^{86,87,92}). Despite this progress, run-time scaling remains a significant obstacle to overcome before ADAPT-VQEs can perform useful computations on real hardware. A balance must be found between run-time and the acceptable level of statistical noise. This is complicated by the combination of gate errors, measurement errors and statistical errors, which may affect VQEs adversely in a non-trivial way. We leave this as an open problem for the community as, in this work, we focus on the noise resilience of ADAPT-VQEs.

This work numerically investigated the maximally allowed gate-error probability _{c} required to achieve chemically accurate predictions as a core metric of VQE performance. Similar to a fault-tolerance threshold in error correction, _{c} should provide a transparent metric to compare the noise resilience of VQEs as well as provide useful guidance for the experimental community. Having demonstrated that _{c} is between 10^{−4} and 10^{−6} for very small molecules (and worse for larger molecules), we conclude that quantum advantage in VQE-based quantum chemistry requires: (i) Substantially improved error mitigation, (ii) error correction, and/or (iii) significantly improved hardware in which gate errors are reduced by orders of magnitude.

We thank Hugo V. Lepage, Flavio Salvati, Frederico Martins, Joseph G. Smith, Wilfred Salmon, and the Hitachi QI team for useful discussions. NM and DRMAS acknowledge useful discussions with Tatsuya Tomaru, Saki Tanaka and Ryo Nagai.

D.R.M.A.-S. and N.M. designed and supervised the project. Y.S.Y., C.K.L. and K.D. wrote the simulation software. K.D. implemented the project and all numerical calculations in close collaboration with C.K.L.. C.K.L. and N.M. developed the noise sensitivity analysis. All authors engaged in technical discussion and the writing of the manuscript.

Data generated during the study is available upon request (E-mail: kd437@cantab.ac.uk or ckl45@cam.ac.uk).

The code used for the simulations and analysis carried out for this work is available upon request from the authors (E-mail: kd437@cantab.ac.uk or ckl45@cam.ac.uk).

The authors declare no competing interests.

Supplementary Information

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