## Abstract

Variational quantum algorithms, a representative class of modern quantum algorithms, provide practical uses of near-term quantum processors. The size of the problem that can be encoded and solved on a quantum processor is limited by the dimension of the Hilbert space associated with the processor. One common approach for increasing the system dimension is to utilize a larger number of quantum systems. Here, we adopt an alternative approach to utilize multiple degrees of freedom of individual quantum systems to experimentally resource-efficiently increase the Hilbert space. We report experimental implementation of the variational quantum eigensolver (VQE) using four-dimensional photonic quantum states of single photons. The four-dimensional quantum states are implemented by utilizing polarization and path degrees of freedom of a single photon. Our photonic VQE is equipped with a quantum error mitigation protocol that efficiently reduces the effects of Pauli noise in the quantum processing unit. We apply our photonic VQE to estimate the ground state energy of the ${\rm{He}} - {{\rm{H}}^ +}$ cation. Simulation and experimental results demonstrate that our experimental resource-efficient photonic VQE can accurately estimate the bond dissociation curve, even in the presence of large noise in the quantum processing unit. We also discuss further possible resource-efficient enhancement of the Hilbert space in photonic quantum processors. Our results propose that photonic systems utilizing multiple degrees of freedom can provide a resource-efficient avenue to implement practical near-term quantum processors.

© 2022 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

## 1. INTRODUCTION

Ongoing efforts to build a useful quantum computer are currently in the noisy intermediate-scale quantum (NISQ) era, characterized by hardware with tens of qubits, noise in the evolution, and no error correction [1]. On the hardware side, several proposals based on different hardware architectures are being pursued and actively developed, e.g., superconducting, trapped ions, and photonic systems [2–4]. At the same time, quantum–classical hybrid algorithms—called variational quantum algorithms (VQAs)—are emerging as promising candidates for near-term practical use of quantum processors. VQAs have applications in a wide variety of fields ranging from chemistry to physics and machine learning [5–14]. VQAs operate by preparing a parameterized trial state on the quantum processor and evaluating a cost function of interest by measuring the state. Then, the cost function is iteratively optimized by varying the parameters of the trial state on a classical computer followed by its preparation and measurement on the quantum processor. One example of VQAs is the variational quantum eigensolver (VQE) where the cost function of interest is the expectation value of a Hermitian operator $H$. By minimizing this cost function, one may obtain the lowest eigenvalue of $H$ [5]. The key advantage of VQE is the ability of solving exponentially large $H$ by the linearly increasing number of qubits in the quantum processor. Since the first demonstration of VQE was reported using a photonic system [5], various achievements in VQE have been reported, e.g., demonstration on different platforms [6–10] and introduction of quantum error mitigation (QEM) [15,16].

Photonic systems have played an essential role in quantum information processing [17–20]. One attractive feature of photonic platforms is the possibility of utilizing multiple degrees of freedom, e.g., polarization, and spatial and temporal modes, of individual photons to encode multiple qubits on a single photon [21,22]. Thus, we can increase the dimension of the Hilbert space without increasing the number of photons in the quantum processor. Note also that some two-qubit gate operations can be easily implemented by addressing qubits to multiple degrees of freedom of a single photon [21].

In this paper, we implement VQE using single-photon four-dimensional quantum states or photonic ququarts. The photonic ququart is demonstrated with polarization and spatial degrees of freedom. The two-qubit Hamiltonian of ${\rm{He}} - {{\rm{H}}^ +}$ is encoded with the single-photon ququart, and its ground state energy is obtained via VQE. Our photonic VQE is also accompanied by the QEM scheme, which remedies the errors introduced by Pauli noise on both degrees of freedom. The experimental efficacy of the QEM scheme is verified with the existence of depolarizing noise with different strengths. Our results demonstrate the usefulness of high-dimensional photonic quantum states in implementing quantum–classical hybrid algorithms.

## 2. THEORY

In this section, we introduce the VQE procedure and discuss possible sources of noise and errors in both classical and quantum processing. We also discuss a QEM scheme that is capable of mitigating the effects of Pauli noise in the VQE.

#### A. Notations

Let ${{\cal H}_d}$ be the $d$-dimensional Hilbert space. A $d$-dimensional pure state $|\psi \rangle$ is a $d \times 1$ column vector in ${{\cal H}_d}$ with the normalization condition $| {|\psi \rangle} | = 1$, where $| \cdot |$ is the vector (${\ell _2}$) norm. We denote the set of density operators on ${{\cal H}_d}$, i.e., $d \times d$ positive semidefinite operators with unit trace, as ${\cal D}({{{\cal H}_d}})$. By a slight abuse of notation, we sometimes denote the density operator of a pure state $|\psi \rangle$ by its label alone without the ket, i.e., $\psi = |\psi \rangle \langle \psi |$. The set of Hermitian operators, i.e, $H = {H^\dagger}$, where ${(\cdot)^\dagger}$ is the conjugate transpose, is denoted by ${\cal S}({{{\cal H}_d}})$. We omit the domains of these operators when it is clear from the context. The expectation value of an observable $H$ with respect to a state $\rho$ is denoted by ${\langle H\rangle _\rho}$, which is defined as ${\langle H\rangle _\rho} = {\rm{tr}}({H\rho})$. The expectation value of $H$ with respect to a pure state $|\psi \rangle$ is ${\langle H\rangle _\psi} = \langle \psi |H|\psi \rangle$. We denote the eigenvalues of $H \in {\cal S}({{{\cal H}_d}})$ in nondecreasing order by ${E_\ell}$, $\ell \in \{{0,1, \cdots ,d - 1} \}$. Then, ${E_0}$ is the smallest eigenvalue of $H$. We use the convention of denoting the identity operator on ${{\cal H}_2}$ by ${\sigma _0}$, and Pauli $X,Y$, and $Z$ operators by ${\sigma _1},{\sigma _2}$, and ${\sigma _3}$, respectively.

#### B. Variational Quantum Eigensolver

VQAs have emerged in the recent times as one of the leading candidates for providing application-oriented quantum computational advantage. At the heart of VQAs is the calculus of variations, i.e., introduction of small perturbations in the system in an attempt to find the maximum or the minimum of some property of interest.

In this work, we are interested in the VQE. The VQE is a special case of VQAs to estimate the eigenvalues of a given Hermitian operator. The given Hermitian operator $H \in {\cal S}({{{\cal H}_d}})$ may correspond to the electronic structure of some molecule of interest, then the lowest eigenvalue of $H$ corresponds to the ground state energy of this molecule. In particular, VQE attempts to estimate ${E_0}$ by minimizing the Rayleigh quotient

The measurement of expectation ${\langle H\rangle _\psi}$ is carried out by measuring a set of appropriate Pauli operators. It is known that any multi-qubit Hamiltonian can be decomposed into a number of Pauli operators with some weight coefficients, i.e.,

with ${w_j} \in R$, and ${\sigma _j}$ are the Pauli strings, i.e., tensor products of multiple Pauli operators. Then, using the linearity of the expectation values, we can obtainThat is, we can measure the expectation values of Pauli strings, and then the appropriately weighted sum of these Pauli expectations gives an estimate on the expectation value of $H$.

Here, we use the VQE to estimate the ground state energy of the ${\rm{He}} - {{\rm{H}}^ +}$ cation [5]. The simplest form of the Hamiltonian of this cation can be represented on two qubits, or on a single ququart, as in our case. The Pauli decomposition of this Hamiltonian has nine terms, some of which can be measured simultaneously by nondegenerate projective measurements on ${{\cal H}_4}$. Consequently, only four Pauli projective measurements are needed to estimate the expectation of our Hamiltonian of interest. See Supplement 1, Section 1 for the detailed Hamiltonian.

Finally, to minimize the expectation value, the target state is parameterized with a parameter set $\vec \theta \in {R^m}$, i.e., $|\psi \rangle = |\psi (\vec \theta)\rangle$. Ideally, $m$ is linear, or at most, polynomial in the number of qubits. At the start of the algorithm, the parameterized quantum state with random values assigned to these parameters is prepared. The expectation ${\langle H\rangle _{\psi (\vec \theta)}}$ is estimated from the above procedure. The estimated value of this expectation is fed to a classical optimizer, which calculates a new set of parameters $\vec \theta$. A new quantum state with these new parameters is prepared, and a new estimate on the expectation is calculated with these parameters. This process is repeated several times until the estimated expectation value converges to a certain value. The minimum expectation value occurring during the above procedure is the estimate ${E_0}$. Figure 1 depicts the working procedure of a VQE including the QEM procedure, which will be discussed later. Note that an ideal VQE does not require QEM.

Ideally, VQE is able to accurately estimate the lowest eigenvalue ${E_0}$ of the Hamiltonian of interest $H$. On the other hand, practical implementations of the VQE may suffer from several non-idealities and noises. In the following, we discuss these sources of noise and the deviation of practical VQE implementations from ideal ones.

#### C. Noise and Errors in VQE

In this section, we outline different kinds of noise and errors in the implementation of VQE. In general, quantum error correction (QEC) is the standard approach for correcting errors in the implementation of quantum algorithms and protocols [23]. However, since the VQAs operate in the paradigm of NISQ computing, we may not have enough resources to implement full QEC. Furthermore, due to the different nature of VQAs from general quantum algorithms, the encountered errors are also of a different nature. We discuss these errors and approaches to remedy these errors in the following.

### 1. Statistical Noise

The estimation of Pauli expectations is achieved by preparing $|\psi \rangle$ and measuring it in the eigenbasis of the corresponding Pauli string $M$ times. Let ${s_j} = {\langle {\sigma _j}\rangle _\psi}$ and ${\hat s_j}$ be its estimate. Then, the difference ${\epsilon _j} = {s_j} - {\hat s_j}$ is the error in the expectation estimation. Since $M$ is finite, statistical noise will contribute to this error. Barring other sources of errors, it is intuitive to think that this error can be reduced by choosing a large $M$. We make this notion more precise in the following by applying Hoeffding’s inequality [24].

Since Pauli strings ${\sigma _j}$ are products of Pauli operators, their spectrum is degenerate, i.e., it has only two eigenvalues ${-}{{1}}$ and ${+}{{1}}$. Then, without loss of generality, we can assume $M$ measurement outcomes ${X_1},{X_2}, \cdots ,{X_M}$ to be independent and identically distributed Bernoulli random variables taking values from $\{{- 1,1} \}$. Then,

and we can apply the Hoeffding’s inequality to precisely bound the probability that the error is greater than some positive number $t$ asThat is, most of the estimated results are concentrated around the true expectation value, and the probability of deviating by $t$ is exponentially small in $M{t^2}$. Therefore, increasing the number of measurements $M$ indeed exponentially decreases the probability of error being larger than some number $t$.

Since the expectation $\langle H\rangle$ is the weighted sum of the expectation of the Pauli strings, we can deduce that its estimates are also going to be concentrated around its true value. However, obtaining a precise bound on its deviation is more involved due to the dependence of different Pauli expectations on each other and is beyond the scope of this paper. We remark only that increasing the number of measurements for the expectation of each Pauli string reduces the statistical errors and improves the estimates of $\langle H\rangle$.

In our experiments, we use $M \approx 4,000$ measurements per Pauli string. This leaves us with a probability of less than 0.013 that the error in estimating ${\langle {\sigma _j}\rangle _\psi}$ is greater than 0.05.

### 2. Optimization Errors

Another source of possible errors in a VQE implementation can be optimization errors. We define optimization errors to be errors where the classical optimizer returns a minimum of the objective function that is not the true minimum. These errors can be further divided into two separate categories: (i) inherent problems of the optimizer, and (ii) optimization errors due to the presence of local minima or vanishing gradients in the landscape of the objective function. We discuss both of these categories in some detail in the following.

In general implementations of VQAs, the cost function is not analytically obtained, which makes it difficult to utilize some gradient-based optimizers. There are a few studies that utilize gradient-based optimizers for VQAs with some success [25,26]. However, these methods suffer from vanishing gradients and generally perform worse than gradient-free optimization methods due to noise [27,28]. Therefore, VQAs often employ gradient-free optimizers including Nelder–Mead, Powell, and constrained optimization by linear approximation (COBYLA). A comparison of various gradient-based and gradient-free optimizers for solving the ground-state energy with VQE can be found in Refs. [29,30].

One problem with these heuristics-based optimizers is that they cannot guarantee the closeness of the obtained solution from the actual solution. Then, one has to carefully choose the stopping criterion for the optimizer. This stopping criterion can be in the form of total number of iterations, some convergence behavior of the objective function values, or a combination of both.

One can only hope to obtain a good quality solution by allowing a large number of iterations. However, it is not possible to exactly characterize the number of iterations in terms of the required quality of the solution. Then, the only possibility is to find a sufficient number of iterations by hit-and-trial. For our case, we found $50 \sim 200$ iterations to be sufficient to obtain the minimum of our objective function depending on the optimizer; see Supplement 1, Section 2 for the test results of various optimizers.

The second challenge is the vanishing gradients of the objective function in the space of state parameters. This phenomenon is referred to as barren plateaus in the landscape of objective functions. It can be caused by the nature of the objective function as well as the noise in the quantum hardware [27,28]. It is being actively studied, and new techniques are being developed to understand and reduce its effects in optimizing parameterized quantum circuits. Some possible remedies to avoid barren plateaus include appropriately modifying the cost function, detecting the existence of barren plateaus using quantum control theory, and trading off the size of the solution space in favor of convergence by reducing the expressibility of ansatz states [31,32].

### 3. Quantum Noise

Quantum noise is the noise present in the state preparation, evolution, and measurement, i.e, anywhere in the quantum processing unit (QPU). Traditionally, readout noise is treated differently from the noise in state preparation and state evolution. However, as we show below, the noise present in our system can be “absorbed” in the readout errors. Consequently, we are able to employ a unified error mitigation scheme for mitigating all types of noise in our QPU.

A quantum channel ${\cal N}$, i.e., a trace-preserving completely positive (TPCP) map, can be represented in terms of its Kraus operators with an input state $\rho$ as

where ${K_j}$ are the Kraus operators satisfying $\sum\nolimits_j K_j^\dagger {K_j} = I$, where $I$ is the identity matrix. We use photonic four-dimensional states in our experiments, which consist of single-photon polarization and path degrees of freedom. The most common quantum noise in this system is depolarization and dephasing noise. These noise types are special cases of Pauli channel noise. This allows us to use a recent QEM scheme proposed for mitigating errors introduced by Pauli and generalized Pauli channels [33]. A Pauli channel is a random unitary channel with the following operator-sum representation: where ${K_j}$ are the Pauli matricesHere, ${p_j}$ is the probability, i.e., ${p_j} \ge 0$ for all $j$, and $\sum\nolimits_j {p_j} = 1$. We can model the Pauli channel noise on two degrees of freedom as

Readout errors in the measurement of a quantum state are modeled as a left stochastic matrix [34]. By letting $p$ and $q$ be, respectively, the ideal and measured (noisy) probabilities obtained after measuring some state $\rho$, we have

where $\Lambda$ is the left stochastic matrix, i.e., ${\Lambda _{j,k}} \in [{0,1}]$, $\sum\nolimits_j {\Lambda _{j,k}} = 1$, which models the behavior of a noisy measurement device [34]. To see the effect of Pauli noise on Pauli measurements, consider the following scenario [33]. Let us assume that we are interested in measuring the Pauli string $P$ on a quantum state $\rho$. We can decompose $\rho$ in the eigenbasis of $P$ asSince the $n$-qubit Pauli channel is a random unitary channel, its effect on $\rho$ is to randomly apply one of the Pauli strings on $\rho$. Furthermore, there does not exist any Pauli string ${P_j}$ such that ${P_j}|{\phi _i}\rangle \langle {\phi _k}|P_j^\dagger = |{\phi _x}\rangle \langle {\phi _x}|$ for any $j,i \ne k$, and $x$. Therefore, the off-diagonal terms remain irrelevant even after passing through a Pauli channel, and we can assume $\rho \approx \tilde \rho = \sum\nolimits_j {\alpha _{j,j}}|{\phi _j}\rangle \langle {\phi _j}|$.

It was shown in Refs. [35,36] that the effect of a discrete Weyl channel (a generalization of Pauli qubit channels) on the eigenstates of a Weyl operator can be modeled as a classical symmetric channel. These results were later generalized for generalized Pauli channels [37]. Since the state of interest $\tilde \rho$ is diagonal in the eigenbasis of $P$, the same argument of modeling the Pauli channel as a classical symmetric channel holds here. The transition matrix of a classical symmetric channel is doubly stochastic. Then, the vector of measurement probabilities with an ideal detector after the effect of the Pauli channel is

where $\Delta$ is the doubly stochastic matrix representing the effect of the Pauli channel. Finally, the effects of measurement errors on ${p^{{\rm{noisy}}}}$ can be accounted for by substituting $p$ in Eq. (10) with the last equation, i.e., where we have defined $\Gamma = \Lambda \Delta$. Finally, we recall that a doubly stochastic matrix is also left stochastic, and the product of two left stochastic matrices is again a left stochastic matrix. Hence, $\Gamma$ is also a left stochastic matrix, and Eq. (13) has the exact same form as that of a noisy detector characterized by $\Gamma$ with a noiseless state.This formulation allows us to treat the errors from the noisy evolution as well as from the measurement noise in a unified manner as measurement errors alone. Consequently, we need to perform only quantum detector tomography, which is equivalent to Pauli channel tomography in this formulation, to obtain ${\Gamma _j}$ for each Pauli string ${P_j}$ that we want to measure. Then, we can perform QEM based on these ${\Gamma _j}$, which mitigates the errors arising from noisy evolution as well as from the measurement noise [33].

Now let us discuss how one can experimentally construct ${\Gamma _j}$. We begin to prepare eigenstates of ${P_j}$. These eigenstates are allowed to evolve naturally under the system noise, and then the measurement in the eigenbasis of ${P_j}$ is performed. Then, the $(k,\ell)$th entry of ${\Gamma _j}$ is the relative frequency of measuring the $\ell$th eigenstate of ${P_j}$, when the $k$th eigenstate of ${P_j}$ is prepared. Once all ${\Gamma _j}$ corresponding to the ${P_j}$ whose measurement is required for Hamiltonian estimation are obtained, we can perform the QEM by the following two methods [34]. We directly invert ${\Gamma _j}$ to obtain the vector of error mitigated probabilities $p_j^{{\rm{mit}}}$ from the experimentally obtained vector of probabilities $p_j^{{\exp}}$, i.e., $p_j^{{\rm{mit}}} = \Gamma _j^{- 1}p_j^{{\exp}}$. However, this may result in a vector that is not a correct probability vector, i.e., elements may be negative or the sum may be greater than one. In such cases, we can still obtain a valid vector of probabilities by projecting the obtained vector back onto the probability simplex.

We remark that this formulation and mitigation of Pauli channel errors in the framework of measurement errors was recently proposed in Ref. [33], and our work also serves as the first experimental validation of this approach. Moreover, while the original proposal was proposed in the context of parameter estimation of generalized Pauli channels, here, we have shown that this QEM scheme is also applicable to the VQE and other NISQ algorithms where only Pauli measurements are utilized. This formulation requires no additional qubits or quantum resources, except for the estimation of noise matrices ${\{{\Gamma _j}\} _j}$. Furthermore, this method of QEM produces reasonable estimates of the objective function as long as the noise matrices ${\Gamma _j}$ are nonsingular. This condition is satisfied, e.g., when the noise is not maximally depolarizing or completely dephasing.

## 3. EXPERIMENT

Figure 2 presents the overall experimental setup of our photonic VQE using four-dimensional quantum states. The heralded single-photon state is prepared via spontaneous parametric downconversion (SPDC) pumped by femtosecond laser pulses at 390 nm and a single-photon detection at the trigger detector D0; see Fig. 2(a). We have initially generated a Bell state $|{\Phi ^ +}\rangle = \frac{1}{{\sqrt 2}}(|HH\rangle + |VV\rangle)$ using a sandwich beta-barium borate (BBO) crystal, which is specially designed for efficient entangled photon pair generation, where $|H\rangle$ and $|V\rangle$ denote horizontal and vertical polarization states, respectively [38–41]. The polarization state of the heralded single photon is prepared to $|H\rangle$ using a polarizing beam splitter (PBS), and it is sent to the photonic QPU shown in Fig. 2(b).

The incoming single-photon state to the QPU is split into two spatial modes $|a\rangle$ and $|b\rangle$ using sets of half- and quarter-wave plates (HWP, QWP) noted as H1 and Q1, and a polarizing beam displacer (PBD1) that transmits and reflects $|H\rangle$ and $|V\rangle$, repectively. The amplitude and relative phase of spatial modes $|a\rangle$ and $|b\rangle$ can be tuned with H1 and Q1. Then, sets of wave plates, H2, Q2, H3, and Q3, placed at each spatial mode encode the polarization state at each spatial mode, and thus, we can generate arbitrary four-dimensional quantum states of

The projection measurements onto Pauil strings, ${\sigma _j} \otimes {\sigma _k}$, can be performed by sets of wave plates {H4, Q4, H5, Q5, H6, Q6} and PBD2. To compensate for the phase drift between spatial modes $|a\rangle$ and $|b\rangle$, we insert a HWP, ${{\rm{H}}_{\rm{p}}}$, between two QWPs at 45° after the PBD2; see Supplement 1, Section 4 for details [42]. During the experiment, we measure and compensate for phase drift every 10 min. Finally, single photons are detected by avalanche photodiodes D1 and D2, and two-fold coincidences between the trigger detector D0 and D1 or D2, D01, and D02 are registered using a homemade coincidence counting unit (CCU) [43,44]. The optical setup of the VQE experiment is tested with the ability to generate and measure arbitrary four-dimensional quantum states with high purity, $P \gt 0.98$. Examples of generated four-dimensional quantum states are given in Supplement 1, Section 3.

The classical processing unit (CPU) receives the measurement outcomes from QPU and performs QEM, linear calculation, and classical optimization. The QEM procedure proposed in theory can be optionally applied before the linear calculation. The CPU calculates the expectation value of Hamiltonian ${\langle H\rangle _n}$ according to Eq. (3). Then, with the input of $\{{\vec \theta _n},{\langle H\rangle _n}\}$, a classical optimizer updates the input parameters, ${\vec \theta _{n + 1}}$, to find the minimum expectation value $\langle H\rangle$. In our experiment, we alter six angles of wave plates, ${\vec \theta _n} = \{{\rm{H1,Q1,H2,Q2,H3,Q3}}\}$, on the state preparation. The rotation angle error in wave plates is ${ \lt 0.1^ \circ}$, which corresponds to the input parameter tolerance of the classical optimizer.

The performance of VQE is highly dependent on the classical optimizer. To choose the best performing optimizer, we have emulated the performance of various gradient-free and gradient-based optimizers using a classical computer, and summarized the results in Table 1. It is obvious that COBYLA provides the best performance in terms of both success probability ${P_S}$ and number of iterations $\bar N$. We have also verified the performance of different optimizers with real VQE experiments. Therefore, in the following VQE experiment, we have utilized COBYLA for a classical optimizer. The details of the emulation and experimental test results for the classical optimizers are presented in Supplement 1, Section 2.

To verify the performance of our photonic VQE using a single-photon ququart, we run VQE to estimate the ground energy of the ${\rm{He}} - {{\rm{H}}^ +}$ cation without the QEM protocol. Figure 3(a) shows the estimated energy expectation ${\langle H\rangle _n}$ at the interatomic distance $R = 0.9$Å with respect to the number of iterations $n$. During the iteration, the estimated energy expectation converges to the theoretical value presented as a red straight line. Figure 3(b) shows the estimated bound energy of ${\rm{He}} - {{\rm{H}}^ +}$ with respect to the interatomic distance $R$. The red line and circles are the theoretical and experimental values for given interatomic distances. The experimental values and error bars are obtained by taking averages and standard deviations of the five minimum points during a single VQE run. We note that the error bars size is smaller than the markers in Fig. 3(b). It clearly shows that the bound energy estimated by our VQE experiment agrees well with theory. The experimentally obtained minimum ground state energy, ${E_g} = - 2.848 \pm 0.004\;{\rm{MJ}}/{\rm{mol}}$ at $R = 0.9\;{ \mathop {\rm A} \limits^ \circ}$, is close to its theoretical value of ${E_{{\rm{th}}}} = - 2.863\;{\rm{MJ}}/{\rm{mol}}$. To verify the effectiveness of our QEM protocol, we prepared initial states with the depolarization noise as $\rho = (1 - \lambda) \cdot |H\rangle \langle H| + \lambda \cdot \frac{I}{2}$, where $I$ is the $2 \times 2$ identity matrix. The noisy states are prepared by changing the coupling efficiency of the horizontal and vertical photons from the initial entangled photon pairs. In the experiment, it was controlled by a partial PBS (PPBS), which partly transmits a vertical polarization state and fine optical alignment [39]. Before and after the VQE run, we checked the noise strength via Pauli channel tomography, and constructed the error mitigation matrix ${\Gamma _n}$ on each Pauli basis and utilized it for QEM.

Figure 4(a) presents the calculated bound energy at $R = 0.9$Å with respect to the amount of the depolarizing noise $\lambda$. Without the QEM protocol, the calculated bound energy increases as the noise increases. On the other hand, our QEM protocol successfully finds correct results regardless of the amount of noise. Note that, as discussed in theory, our QEM protocol can efficiently remedy noise except for $\lambda = 1$. Figure 4(b) shows the bound energy with respect to the interatomic distance $R$ with noise of $\lambda = 0.20 \pm 0.02$. These experimental results clearly show the effectiveness of our QEM protocol, which successfully finds the correct results without further quantum resources.

## 4. CONCLUSION

We have presented an experimental implementation of VQE on a photonic ququart to estimate the ground state energies of ${\rm{He}} - {{\rm{H}}^ +}$ cations. Using polarization and path degrees of freedom of a single photon, we were able to encode two-qubit Hamiltonians on a single photon. Additionally, we employed the Pauli channel estimation and QEM scheme to reduce the effects of noise in QPU and obtain more accurate estimates of ground state energies.

Our results clearly show that high-dimensional photonic quantum states based on multiple degrees of freedom provide a resource-efficient way to implement VQAs. There are several ways to further expand the Hilbert space by utilizing multiple degrees of freedom. For example, one can increase the number of photons. In our experiment, we have a heralding single photon that can implement another four-dimensional Hilbert pace [45]. One can also increase the number of path degrees of freedom [46]. In principle, the number of paths can be infinite. In experiment, it has been demonstrated to generate and verify two-party entanglement in 32 spatial dimensions, which is comparable to a 10-qubit Hilbert space [46]. Note that each spatial mode can be independently controlled by using linear optical elements, so various quantum states can be implemented. It is also remarkable that it significantly reduces the experimental resources by employing 2D beam displacers. By adding polarization degrees of freedom in this scheme, we can efficiently increase the size of the Hilbert space. We also note that time-bin encoding using a loop-based architecture can also provide an experimental resource-efficient way to increase the Hilbert space [47–50]. Finally, photonic systems utilizing more than two degrees of freedom, e.g., polarization-path-orbital angular momentum, can be an interesting approach to provide a scalable way to expand the Hilbert space [22]. It is also notable that many high-dimensional photonic quantum systems based on multiple degrees of freedom can initially prepare entangled states [22,45,46] that can effectively reduce the required circuit depth to implement particular ansatz states.

We remark that the experimental demands of some entangling operations in a multiple degrees of freedom encoding system are comparable to those of single-qubit operation. For instance, in our photonic system, the experimental difficulty of Bell state measurement, which projects the input quantum states to four Bell states, is comparable to simple Pauli string projection measurement. This feature suggests our system is a natural choice for a recently proposed scheme to reduce the number of measurement settings via entangling measurements [51].

## Funding

Korea Institute of Science and Technology (2E31021); MSIP/IITP (2020-0-00972, 2020-0-00947, 2021-0-02046); National Research Foundation of Korea (2019M3E4A1078662, 2019M3E4A1079777, 2019R1A2C2006381, 2019R1A2C2007037, 2019R1I1A1A01059964, 2021M1A2A2043892, 2021R1C1C1003625).

## Disclosures

The authors declare no conflicts of interest.

## Data availability

All relevant data are available from the corresponding author upon request.

## Supplemental document

See Supplement 1 for supporting content.

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