Few-shot estimation of entanglement with Bell measurement assistance

Abstract

Entanglement is fundamental to quantum physics and information processing. In this work, we introduce the Few-Shot Randomized Measurement (FSRM) method, developing an unbiased estimator for mixed-state entanglement from just three experimental shot outcomes. By incorporating the Bell measurement (BM), we supplement the traditional computational-basis measurement to enhance the randomized measurement scheme, which is scalable to n-qubit systems via BMs on qubit pairs. Our approach enables direct estimation of entanglement through random unitary evolution in a photonic system. Compared to the classical shadow method, BM-enhanced FSRM requires no prior knowledge of the local unitaries, offering greater robustness against unitary imperfections. Additionally, we find that utilizing more versatile measurement settings with fewer repeats per setting is more efficient under fixed measurement resources. Our protocol and experimental demonstration represent a significant advancement in the efficient and practical characterization of quantum states.

Introduction

Entanglement¹, as the very nature of nonlocal correlation, plays a fundamental role in physics, from quantum foundations to quantum many-body physics², and is also a central fuel for quantum information processing³. Characterization and quantification of entanglement is one of the core topics in the field and is challenging in general^1,4,5. Methods like the entanglement witness^4,5,6 need the prior knowledge of the state form, and the full tomography method not only needs extensive measurement resources but also may introduce biasedness in the entanglement calculation⁷.

The difficulties lie in the essence that the boundary between the entangled states and the separable states is very complicated^8,9,10,11. In this way, entanglement measures used to quantify the amount of entanglement are thus highly nonlinear and complex^1,8,12, and cannot be directly evaluated. One widely employed entanglement measure is the entanglement negativity ${\mathcal{N}}={\sum }_{\lambda < 0}| \lambda |$^13,14, with λ the eigenvalues of ${\rho }_{AB}^{{T}_{B}}$, which is obtained by the partial transpose (PT) operation T_B on the joint density matrix ρ_AB. Such entanglement measures are inherently highly nonlinear and complex^1,8, making their direct evaluation intractable in general.

To overcome this, recent work has focused on low-order approximations of entanglement negativity based on PT moments^15,16. These are quantities of the form ${p}_{k}=Tr[{({\rho }_{AB}^{{T}_{B}})}^{k}]$^17,18, with ρ_AB as quantum state with two subsystems A and B, which can already detect entanglement at small k. For instance, the p₃-PPT criterion¹⁹ uses ${{\mathcal{N}}}_{3}={p}_{2}^{2}-{p}_{3}$, where ${{\mathcal{N}}}_{3} > 0$ certifies entanglement in mixed states. Yet, conventional approaches to PT moments require access to k identical copies of the state simultaneously (e.g., k = 3 for p₃), posing significant practical hurdles.

Recently, randomized measurement (RM) schemes^20,21 have emerged as powerful tools for directly estimating nonlinear state functions without multi-copy control^{18,19,22,23,24,25,26,27,28}. In a standard RM protocol, the measurement procedure consists of N_U rounds. In the i-th round, a random unitary U_i is applied, followed by a fixed measurement ${\mathbb{M}}$, which effectively corresponds to measuring in the rotated basis ${U}_{i}^{\dagger }{\mathbb{M}}{U}_{i}$. In each round, it repeats N_M times and generates N_M n-bit binary shot outcomes (e.g., 010...1) for an n-qubit system. For sufficiently large N_M, one can estimate the conditional probabilities (b∣U) for all bit strings b. By averaging statistical moments over the unitary ensemble, such as ${{\mathbb{E}}}_{U}[{(({\bf{b}}| U))}^{k}]$, it is then possible to estimate some k-th order nonlinear functions of the density matrix, ${{\mathcal{T}}}_{k}(\rho )=Tr({T}_{k}{\rho }^{\otimes k})$. These include, in particular, the PT moments p_k. Specifically, the k-th order PT moment for subsystems A and B is given by ${p}_{k}=Tr[({W}_{\to }^{A}\otimes {W}_{\leftarrow }^{B}){\rho }^{\otimes k}]$, where W_→/← are (anti-)clockwise cyclic permutation operators on subsystem A/B^18,19. In this case, ${T}_{k}={W}_{\to }^{A}\otimes {W}_{\leftarrow }^{B}$. Importantly, RM exploits the statistical properties of random unitaries to access such observables without requiring explicit knowledge of the sampled unitaries²⁹.

A related breakthrough is the classical shadow (CS) method^30,31,32. Unlike RM, which estimates probabilities by collecting many outcomes per unitary, CS typically acquires only one shot outcome on each measurement round, i.e. N_M = 1. While one shot outcome per unitary precludes direct probability estimation, with the knowledge of U_i, it enables construction of a classical shadow snapshot: ${\widehat{\rho }}_{i}={{\mathcal{M}}}^{-1}[{U}_{i}^{\dagger }\left|{{\bf{b}}}_{i}\right\rangle \left\langle {{\bf{b}}}_{i}\right|{U}_{i}]$, where ${{\bf{b}}}_{i}\in {{\mathbb{F}}}_{2}^{n}={\{0,1\}}^{n}$ is the observed n-bit shot outcome at the i-th round measurement for an n-qubit state. k independent shadows are needed to calculate an unbiased CS estimator for PT-moment as $\widehat{{p}_{k}}=Tr[({W}_{\to }^{A}\otimes {W}_{\leftarrow }^{B}){\widehat{\rho }}_{{l}_{1}}\otimes \cdots \otimes {\widehat{\rho }}_{{l}_{k}}]$. Thus, the estimation becomes:

$$\left\{\begin{array}{l}{p}_{2}={\mathbb{E}}\left[\widehat{{p}_{2}}({{\bf{b}}}_{i},{{\bf{b}}}_{j};{U}_{i},{U}_{j})\right],\\ {p}_{3}={\mathbb{E}}\left[\widehat{{p}_{3}}({{\bf{b}}}_{i},{{\bf{b}}}_{j},{{\bf{b}}}_{l};{U}_{i},{U}_{j},{U}_{l})\right],\end{array}\right.\,(\,\mathrm{CS})$$

(1)

where ${\mathbb{E}}$ is the expectation over experiments with random unitaries, and with one-shot outcome collected for each unitary, i.e., by averaging over all pairs (i, j) (for p₂) or triples (i, j, l) (for p₃) of measurement rounds. This demonstrates that CS requires only two and three shot outcomes to calculate $\widehat{{p}_{2}}$ and $\widehat{{p}_{3}}$, respectively. Hence, CS intrinsically exhibits the few-shot property, which enhances the efficiency in probing the mixed-state entanglement¹⁹. However, the explicit dependence of ${\widehat{\rho }}_{i}$ on the specific U_i makes CS estimators vulnerable to unitary implementation errors, which can cause a mismatch between the assumed and actual unitaries and introduce bias and error into the estimates.

In this work, we introduce an efficient and robust way for the estimation denoted as few-shot randomized measurements (FSRM), a hybrid approach combining RM’s noise resilience with CS’s few-shot efficiency, as shown in Fig. 1a. FSRM, on the one hand, does not require the precise information of the random unitary transformation; on the other hand, it estimates p₂ and p₃ using 2 or 3 outcomes per unitary (Fig. 1b). The estimation of p₂ and p₃ in FSRM can be summarized as:

$$\left\{\begin{array}{ll}{p}_{2}={\mathbb{E}}\left[\widehat{{p}_{2}}({{\bf{b}}}_{i,1},{{\bf{b}}}_{i,2})\right],\\ {p}_{3}={\mathbb{E}}[\widehat{{p}_{3}}({{\bf{b}}}_{i,1},{{\bf{b}}}_{i,2},{{\bf{b}}}_{i,3})],\end{array}\right.\,(\,\mathrm{FSRM})$$

(2)

with b_i,1, b_i,2 and b_i,3 are observed shot outcomes obtained from the experiment under same randomized unitary U_i at the i-th measurement round and the expectation ${\mathbb{E}}$ here is for the average over a certain FSRM experimental process. Unlike CS, we also allow FSRM to incorporate Bell-basis measurement (BM) to enhance the estimation of p₃, thus called ‘BM-enhanced FSRM’, as shown in Fig. 1b.

Fig. 1: Demonstration of our protocol.

The connected blue balls are for the n-qubit quantum state, and the boxes are for the random local unitary. The diagrams represent the order of operations. A Demonstration of few-shot randomized measurements (FSRM). In FSRM, the protocol consists of N_U measurement rounds. In the i-th round, the measurement basis is rotated by a randomly chosen unitary, resulting in ${U}_{i}^{\dagger }{\mathbb{M}}{U}_{i}$, where ${\mathbb{M}}$ is a fixed measurement. Unlike standard RM or CS methods, only N_M = k outcomes are collected per round, denoted by b_i,1,...,b_i,k, which are then used to get a real number O(b_i,1,...,b_i,k) from some function O. The average of the real numbers is the estimation of the target quantity ${{\mathcal{T}}}_{k}(\rho )$. In particular, choosing k = 2 or 3 allows estimation of the mixed-state entanglement moments p₂ and p₃. Importantly, FSRM does not require recording the applied unitaries, relying solely on the few collected outcomes to obtain an unbiased estimation. B Demonstration of BM-enhanced FSRM. We slice the state into pairs of qubits. After the random unitary rotation shown in (A), all pairs undergo a Bell-basis measurement (BM) or a computational-basis measurement (CM) with some probability, i.e., ${\mathbb{M}}\in \{{\rm{CM,\; BM}}\}$. For BM, we record bases with labels 0, 1, 2, 3, and for CM, we record bases with labels 00, 01, 10, 11 for post-processing. The overall measurement basis can be represented by an n-bit string $\overrightarrow{s}$. These recorded bases, with the measurement shot outcomes, can give an unbiased estimation following the process in (A).

We experimentally validate both FSRM and CS in biphoton experiments for estimating mixed-state entanglement. Remarkably, due to the irrelevance feature of BM-enhanced FSRM to the information of random unitary, FSRM shows great robustness to the unitary noise compared with CS. That is, FSRM estimators achieve the convergence to the true value, but CS estimators do not, with the same measurement data collected in the experiment. Additionally, we explore improvements in the sampling efficiency by using more diverse measurement settings and reducing the number of shot outcomes per setting within the constraints of quantum resources. Our work enhances the RM scheme for characterizing photonic quantum states, particularly in entanglement quantification, and can potentially be extended to multipartite systems.

Results

BM-enhanced FSRM scheme

The CS scheme employs an effective random unitary channel and its inverse to generate the classical snapshots of a quantum state³⁰, which are then used to calculate the target properties of the state. Alternatively, the random unitary channel can be leveraged differently—by exploiting its statistical properties²³, a technique known as the randomized measurement (RM) scheme²⁹.

Let us first recast the basics of such RM²⁹, focusing on an n-qubit system. The experiment consists of N_U measurement rounds, and for each round N_M experimental outcomes are collected. In each round, the RM scheme samples a random unitary from some ensemble, e.g., local random unitaries ${U}_{i}={\otimes }_{j}{u}_{i}^{(j)}$, where u^(j) is the single-qubit unitary on the j-th qubit, randomly chosen from Haar measure. For each U_i, N_M shot outcomes (normally a big number) are collected to calculate the probabilities (b∣U) on the basis ${\bf{b}}\in {{\mathbb{F}}}_{2}^{n}$. In the experiment, RM needs complete measurement on all bases b If there exists a function ${O}_{k}({{\bf{b}}}_{1},..,{{\bf{b}}}_{k}):{{\mathbb{F}}}_{2}^{nk}\to {\mathbb{R}}$, with (b₁,...b_k) as k-bases pairs, and its corresponding operator ${O}_{k}={\sum }_{{{\bf{b}}}_{1},..,{{\bf{b}}}_{k}}{O}_{k}({{\bf{b}}}_{1},...{{\bf{b}}}_{k})\left|{{\bf{b}}}_{1}\ldots {{\bf{b}}}_{k}\right\rangle \left\langle {{\bf{b}}}_{1}\ldots {{\bf{b}}}_{k}\right|$ satisfies:

$${{\mathbb{E}}}_{U}{U}^{\dagger \otimes k}{O}_{k}{U}^{\otimes k}={T}_{k}$$

(3)

where ${{\mathcal{T}}}_{k}(\rho )=Tr({T}_{k}{\rho }^{\otimes k})$ for a k-th order property of one quantum state and ${{\mathbb{E}}}_{U}$ here is the expectation over all random unitary U_i. Then one can calculate the nonlinear property of ${{\mathcal{T}}}_{k}(\rho )$ by multiplication of conditional probabilities^25,29,

$${{\mathcal{T}}}_{k}(\rho )={{\mathbb{E}}}_{U}\mathop{\sum }\limits_{({{\bf{b}}}_{1},..,{{\bf{b}}}_{k})\in {{\mathbb{F}}}_{2}^{nk}}{O}_{k}({{\bf{b}}}_{1},..,{{\bf{b}}}_{k})\,\mathop{\prod }\limits_{l=1}^{k}\Pr ({{\bf{b}}}_{l}| {U}_{i}).$$

(4)

Here, the left-hand shows the predetermined target ${{\mathcal{T}}}_{k}$, and the right-hand shows the corresponding experimental process to estimate the target. $({{\bf{b}}}_{l}| {U}_{i})=\langle {{\bf{b}}}_{l}| {U}_{i}\rho {U}_{i}^{\dagger }| {{\bf{b}}}_{l}\rangle$ is the observed probability on each measurement round. By transversing all bases and averaging over random unitary, we get the estimation of the k-th order nonlinear property ${{\mathcal{T}}}_{k}(\rho )$ from Eq. (4) like p₂. In practice, to obtain a reasonably accurate (b_l∣U_i), we need to repeat the experiments with U_i many times, and a large N_M is required, which seems to hinder RM in the few-shot scenario.

Inspired by the CS scheme, here we can construct an unbiased estimator of a k-th order function ${{\mathcal{T}}}_{k}$ within the RM framework using few shot outcomes^18,33, and we denote such a modification of RM about the post-processing strategy as few-shot randomized measurements (FSRM). The key idea is that we replace the chosen k-bases pairs (b₁,..,b_k) (independent of U_i) in Eq. (4) into the observed shot outcomes (b_i,1,..,b_i,k) collected in the i-th measurement round with unitary U_i, i.e., let N_M = k. At the time, we didn’t traverse all possible k-bases pairs but collected the exact k shot outcomes to form one k-shot-outcomes pair from the experiments. In this way, we could construct the unbiased estimator as O_k(b_i,1,...,b_i,k) with

$${{\mathcal{T}}}_{k}(\rho )={{\mathbb{E}}}_{U}{{\mathbb{E}}}_{\{{\bf{b}}\}}\left[{O}_{k}({{\bf{b}}}_{i,1},..,{{\bf{b}}}_{i,k})\right],$$

(5)

where O_k is same with that in Eq. (4). The unbiasedness of this estimator is guaranteed by the construction of O_k in Eq.(3) and the statistical independence of the k shot outcomes, as derived in the Methods section. Here, ${{\mathbb{E}}}_{\{{\bf{b}}\}}$ is the expectation over repetitive experiments under same unitary U_i. These two expectations ${{\mathbb{E}}}_{U}$ and ${{\mathbb{E}}}_{\{{\bf{b}}\}}$ can be replaced by one ${\mathbb{E}}$, which is expectation over experiments with a certain FSRM process (see Methods). The whole process of FSRM is summarized in Fig. 1a, where we randomly chose N_U unitaries and for each chosen unitary implemented before measurement, we collect k shot outcomes (N_M = k), and these k shot outcomes help to calculate one estimation for k-th order nonlinear property ${{\mathcal{T}}}_{k}$ with proper O({b_j}). Note that, unlike the estimation of linear observables, we give an unbiased estimation of ${{\mathcal{T}}}_{k}(\rho )$ from every k shot outcomes. For example, we need N_M = 2 to estimate p₂ and N_M = 3 to estimate p₃.

It is also worth mentioning that FSRM enjoys features and advantages like robustness and efficient post-processing from RM, and a few-shot property from CS. A brief comparison among CS, RM, and FSRM schemes is in Table 1, and we leave detailed explanations in the Supplementay Note 1.

Table 1 Comparison of RM schemes

CS, RM, and FSRM schemes would exhibit different performances in the presence of noise. Each protocol essentially consists of two parts: a physical setup for data collection and a classical computation step for processing the target quantity. Here, we assume that all errors originate from the physical setup, while the classical computation is free from additional noise. In practical applications, measurements are often subject to stochastic noise, which is typically modeled as Gaussian noise. Mathematically, this noise manifests through a modification of the unitary rotation in the i-th measurement round, expressed as ${U}_{i}\to {\widetilde{U}}_{i}={e}^{i\epsilon {H}_{i}}{U}_{i}$ with ϵ represents the noise strength, and H_i is a random Hermitian operator³⁴. CS methods rely on the information encoded in the measurement basis, specifically the details of the unitary operation, i.e., U_i. Consequently, under such noise conditions, CS methods fail to provide accurate estimates. As illustrated in Fig. 2, the error in property estimation with CS, RM, and FSRM methods varies notably in the presence of Gaussian noise. The figure reveals that CS introduces a bias directly related to ϵ, limiting its effectiveness under stochastic noise. In contrast, FSRM methods exhibit robustness, consistently converging to the true value despite the noise. A more in-depth discussion of the effects of Gaussian noise on these measurement strategies is provided in Methods and Supplementary Note 3.

Fig. 2: CS methods and RM-like methods under Gaussian noise.

Here, RM-1/2/3 is for using the RM or the FSRM method in estimating the 1/2/3-th order target ${{\mathcal{T}}}_{1/2/3}$. Here, we consider the strength of the Gaussian noises is ϵ = 0.5. Because both RM and FSRM methods essentially utilize the average property of the unitary statistics (k-twirling channel) to extract the information, they share the same performance lines. The y-axis is the difference between the practical channel and the ideal channel, quantifying the error in the estimation. Details are left in Supplementary Note 3.

In the context of mixed-state entanglement evaluation, one needs to estimate the PT moments p₂ and p₃. For the 2-th moment, it can be expressed as ${p}_{2}=tr({\mathbb{S}}{\rho }_{AB}^{\otimes 2})$, with ${\mathbb{S}}$ being the swap operator on 2 copies. It is clear that p₂ equals the purity of ρ and is thus independent of the specific partition A∣B¹⁹. We provide a short proof in methods. Suppose the local random unitary ensemble is applied in RM, one has

$${p}_{2}={\mathbb{E}}[{O}_{2}({{\bf{b}}}_{i,1},{{\bf{b}}}_{i,2})],\,{O}_{2}({{\bf{b}}}_{i,1},{{\bf{b}}}_{i,2})=d{(-2)}^{h({{\bf{b}}}_{i,1},{{\bf{b}}}_{i,2})}$$

(6)

with h(b_i,1, b_i,2) being the Hamming distance between the two observed n-bit shot outcomes^24,29. Here, the expectation is on both the random unitary choice and the measurement result.

For the 3-order one, we follow the post-processing given in ref. ¹⁸, which in general needs a global random unitary or BM between subsystems A and B after independent random unitary evolution on both of them. Indeed, as illustrated in Fig. 1b, for an n-qubit system, BMs on qubit pairs are sufficient to realize the estimation. Here we denote the outcome of BM on $(j,j{\prime} )$ qubit-pair as ${\beta }^{(j,j{\prime} )}=\{0,1,2,3\}$. The specific partition is labeled by an n-bit string $\overrightarrow{a}$, with a_i = 1(0) for qubit-i in A(B), respectively. Additionally, we use another n-bit string $\overrightarrow{s}$ to indicate whether the qubit is involved in BM or not. Of course, $\overrightarrow{s}$ should have even number of ‘1’s, and we randomly sample in total 2ⁿ⁻¹ possible $\overrightarrow{s}$ configurations with equal probability. In short, the corresponding estimator of p₃ is

$$\begin{array}{l}{p}_{3}={\mathbb{E}}{{\mathbb{E}}}_{\overrightarrow{s}}\left[{O}_{3}({{\bf{b}}}_{i,1},{{\bf{b}}}_{i,2},{{\bf{b}}}_{i,3};\overrightarrow{s})\right],\\ {O}_{3}({{\bf{b}}}_{i,1},{{\bf{b}}}_{i,2},{{\bf{b}}}_{i,3};\overrightarrow{s})=\frac{1}{2}{(-1)}^{\overrightarrow{a}\cdot \overrightarrow{s}}\mathop{\prod }\limits_{j:{s}_{j}=1}f(j,j{\prime} )\mathop{\prod }\limits_{j:{s}_{j}=0}g(j).\end{array}$$

(7)

Here, f and g functions are for the BM and CM outcome of qubit-pair $(j,j{\prime} )$ and qubit j, determined by the measurement configuration vector $\overrightarrow{s}$. Here, ${{\mathbb{E}}}_{\overrightarrow{s}}$ means expectation over randomly choose vector $\overrightarrow{s}$ with same probability. And they show respectively

$$\begin{array}{lll}\;\,f(j,j^{\prime})=1-{(-2)}^{\mathrm{wt}({\beta }_{i,1}^{(j,j{\prime} )},{\beta }_{i,2}^{(j,j{\prime} )},{\beta }_{i,3}^{(j,j{\prime} )})}\\{\mathrm{and}}\,g(j)=1+{(-2)}^{[{\mathrm{wt}}({{\bf{b}}}_{i,1}^{(j)},{{\bf{b}}}_{i,2}^{(j)},{{\bf{b}}}_{i,3}^{(j)})-1]},\end{array}$$

(8)

where wt() counts the number of same elements for β and b. More details are left in the Methods.

In the case of 2-qubit, with A and B containing each of them, i.e., $\overrightarrow{a}=10$. The vector $\overrightarrow{s}=\{00,11\}$, which indicates that one conducts CM and BM with equal probability on a 2-qubit system. By Eq. (7), the post-processing is just

$${O}_{3}=\left\{\begin{array}{l}\frac{1}{2}g(1)g(2),\,\overrightarrow{s}=00,\\ -\frac{1}{2}f(1,2),\,\overrightarrow{s}=11.\end{array}\right.$$

(9)

In summary, the procedure for estimating p₂ and p₃ is as follows. We implement random local unitaries ${U}_{i}={\otimes }_{j}{u}_{i}^{(j)}$ on the state ρ and randomly select a measurement setting $\overrightarrow{s}$. For each sampled unitary U_i and $\overrightarrow{s}$, we repeat the process three times to collect measurement outcomes (b_i,1, b_i,2, b_i,3), and finally calculate the function ${O}_{3}({{\bf{b}}}_{i,1},{{\bf{b}}}_{i,2},{{\bf{b}}}_{i,3}| \overrightarrow{s})$ in Eq. (7) as the estimation of p₃. For p₂, we only need the measurement setting all being CM, say $\overrightarrow{s}=\overrightarrow{0}$, and two outcomes are sufficient to obtain O₂(b_i,1, b_i,2) as an estimation. By repeating this process many times, we average the results to obtain the final estimations of p₂ and p₃. The post-processing here is straightforward and time-saving compared to the original RM scheme and CS scheme, which involve calculating multiple shadow snapshots and their multiplications.

Experimental results

In our experiments, states with polarization entanglement are produced through the parameter-down-conversion (PDC) process in the periodically poled potassium titanyl phosphate (ppKTP) crystal within a Sagnac interferometer³⁵. Pure or mixed states could be easily produced by adjusting the pump laser. In the experiments, we produced both a mixed state and a maximally entangled pure state, as detailed in Methods. The produced photons are then sent into two uncharacterized single-mode fibers (SMF) before measurement. These uncharacterized SMFs induce unknown, drifting unitary rotations on the entangled state. Consequently, the actual measurement basis deviates from the intended. We utilize this discrepancy to experimentally simulate a scenario where the measurement basis is unknown and unstable.

The measurement setup is illustrated in Fig. 3, where two photons undergo local random unitaries using Quarter-Half-Quarter (Q-H-Q) waveplates³⁶. Because a BM could be decomposed with a CNOT gate and a Hadamard gate before CM, we can streamline the process of random CM and BM by randomly inserting such gates between qubit pairs and finally conducting CM. In the experiment, two optical switches, each with one input and two outputs, made by motorized mirrors, control the ratio of photon pairs that go through the inner or outer path, ensuring a 50–50% ratio between CM (inner path) and BM (outer path). CM involves measuring horizontal and vertical polarizations in our experiments, resulting in a string of {0, 1}ⁿ. On the inner path, an additional all-optical CNOT gate consisting of three partially-polarized-beam-splitters (PPBSs)^37,38,39, and a Hadamard gate on the control part were implemented.

Fig. 3: Sketch of the measurement setup.

Entangled biphoton states are transmitted to the measurement setup through uncharacterized fibers. Each photon within the setup undergoes a randomized unitary, selected from a Circular Unitary Ensemble (CUE) implemented using a set of quarter-half-quarter waveplates (depicted by the orange circle). Two optical switches, each featuring one input and two outputs, determine whether the pair of photons will be measured in computational basis (CM) along the outer path or in Bell basis (BM) along the inner path, with a probability of 1/2. Along the outer path, the CM setup consists of two polarizing beam splitters (PBSs). In contrast, the inner path includes an all-optical CNOT gate (depicted by the green circle), which consists of three Partial-Polarization-Beam-Splitter (PPBSs) with t_H = 1 and t_V = 1/3, following the setup in refs. ^37,38,39. This setup, combined with an additional Hadamard gate (22.5^circ half-wave plate) and CM, forms a complete BM. Finally, single-photon detectors (SPDs) register the pair of photons. Further details can be found in the Method.

In the experiment, we collect 3 shot outcomes for each measurement setting and unitary. For photon pairs through the outer path, i.e., CM, we select 2 shot outcomes to calculate O₂(b₁, b₂) as the estimation of p₂. By repeating this process numerous times, we accurately estimate p₂. The results for p₂ are depicted in Fig. 4a. Additionally, we select 1 shot from the shot outcomes using CM per unitary and combine the unitary information to calculate the classical shadow of the state ${\widehat{\rho }}_{i}{=\bigotimes }_{j}(3{u}_{i}^{(j)}|{{\bf{b}}}_{i}^{(j)}\rangle \langle {{\bf{b}}}_{i}^{(j)}|{u}_{i}^{(j)\dagger }-{{\mathbb{I}}}_{2})$³⁰, j is for the j-th qubit and the shot outcomes is $\left|{{\bf{b}}}_{i}\right\rangle {=\bigotimes}_{j}|{{\bf{b}}}_{i}^{(j)}\rangle$, and i is for the i-th round experiment with unitary ${U}_{i}{=\bigotimes }_{j}{u}_{i}^{(j)}$ and ${u}_{i}^{(j)}$ is randomly chosen from Haar measure, which is used to estimate p₂ and p₃ from Eq.(1). Results for CS are presented in Fig. 4b, d.

Fig. 4: Experimental results comparing the prediction accuracy for p₂ and p₃ between the BM-enhanced FSRM and CS schemes.

The red dashed line indicates the true value, while the solid lines represent 10 independent experimental trials. The figures demonstrate how the estimates stabilize as more experimental data are incorporated, where N denotes the total number of collected photon pairs. A and B show the estimation of p₂ using the BM-enhanced FSRM method, while panel C shows the estimation of p₂ using the CS method. D and E depict the estimation of p₃ using the BM-enhanced FSRM method, with F showing the estimation of p₃ using the CS method. For the BM-enhanced FSRM method, p₂ is estimated using 2-shot outcomes per unitary, such that N = N_U × 2, where N_U is the number of unitaries applied. p₃ is estimated using 3 measurement outcomes per unitary (N = N_U × 3). In the estimation of p₃, an additional error mitigation technique is applied to correct errors induced by imperfect CNOT gate for the fixed Bell measurement. Each experiment is independently repeated ten times, represented by different lines. For CS, only one shot per unitary is required for prediction. By comparing the results, we observe that although both schemes utilize the same dataset from the experiment, BM-enhanced FSRM shows convergence to the true values, while the CS scheme exhibits bias in practice.

By employing 3 shot outcomes in both the outer and inner paths, we estimate p₃ with the FSRM scheme by calculating ${O}_{3}({{\bf{b}}}_{1},{{\bf{b}}}_{2},{{\bf{b}}}_{3}| \overrightarrow{s})$ in Eq. (7). It is worth noting that randomized measurements correspond to measurement settings of the form ${U}_{i}{\mathbb{M}}{U}_{i}^{\dagger }$ with many different U_i. The FSRM method is naturally robust against noise in the random unitaries U_i, which is difficult to calibrate. Moreover, for imperfect fixed measurements ${\mathbb{M}}$, it remains compatible with error-mitigation techniques based on calibrating the fixed measurement. BM-enhanced FSRM, here, is likely to be affected by imperfect CZ gates. And we implement error mitigation techniques. The intuition comes from the fact that the randomized measurement provides a perfect equivalent permutation operator from noisy measurements by updating the post-processing. To mitigate measurement imperfections of BM, we utilize the information from measurement tomography⁴⁰ to update the function ${O}_{3}({{\bf{b}}}_{1},{{\bf{b}}}_{2},{{\bf{b}}}_{3}| \overrightarrow{s})$ in Eq. (5) by replacing the corresponding Bell basis of b_i with the real measurement M_i. Despite the practical fidelity of BM being less than 80%, we can make the updated inaccuracy negligible, being around 10⁻⁸, and proceed to estimate the true value of p₃. Further details on error amendment can be found in Supplementary Note 2.

Results from 10 independent experiments for a mixed state are depicted in Fig. 4. The tomography results indicate the 2nd PT moment as p₂ = 0.858 and its 3rd one as p₃ = 0.3580. Utilizing the FSRM scheme and predicting from N_U = 10⁴ response numbers, we obtain experimental estimations of $\widehat{{p}_{2}}=0.826\pm 0.009$ and $\widehat{{p}_{3}}=0.344\pm 0.034$ (see Fig. 4a, b, d, e), converging to the true state values. Additionally, we estimate the mixed state entanglement as ${{\mathcal{N}}}_{3}={p}_{2}^{2}-{p}_{3}=0.338\pm 0.019$¹⁹. The obtained entanglement value exceeds 0, affirming the existence of mixed-state entanglement. The uncertainties in the estimations stem from 10 independent experiments. However, employing (one-shot) CS scheme using the same dataset as the FSRM scheme, we estimate $\widehat{{p}_{2}}=0.593\pm 0.008$ and $\widehat{{p}_{3}}=0.229\pm 0.010$ (see Fig. 4c, f). The results in Fig. 4a, b, d, e validates our FSRM methods. And, the data in Fig. 4a–c comes from the same data without any mitigation, and the only difference is CS scheme (b) utilizes the information of unitary, while the FSRM method (a and b) does not. Our analysis shows that the derivation of predictions of the CS scheme may come from the need for the exact information of the applied random unitaries, and thus the unitary errors can affect much, whereas for the FSRM scheme, such kind of information is unnecessary. This discrepancy between CS and FSRM schemes aligns with our analysis regarding Gaussian noise in unitaries, as detailed in Supplementary Note 3. Consequently, FSRM proves to be more robust for noisy and practical devices.

Finally, we investigate the statistical error of the FSRM scheme by taking p₂ of a two-photon state as an example. The total measurement time is N = N_U × N_M, i.e., the multiplication of rounds and shot number. The impact of N_M and N on the convergence of the statistical error is illustrated in Fig. 5, in which each line represents the results of over 100 independent experiments. For FSRM, one can get an estimation of every 2-shot from N_M-shot under the same unitary, and finally, the result is averaged for all N_U unitaries. For RM, we utilize all N_M-shots under the same unitary to compute the conditional probability (b∣U), which becomes equivalent to our FSRM scheme as N_M tends to be large enough, as shown in Fig. 5 for N_M = 200. Notably, if N is fixed, a smaller N_M, i.e., more diversifying unitaries, yields better performance. Through linear regression, we ascertain $\log {\rm{Err}}=-0.5\log N+0.4393$ as N_M = 2, and for RM (N_M = 200), $\log {\rm{Err}}=-0.5\log N+0.8726$, which is labeled by a cross mark in Fig. 5. Consequently, utilizing FSRM results in a 4.3 dB improvement in estimation compared to the original RM.

Fig. 5: Experimental results of the statistical error performance in the prediction of p₂ with FSRM.

N for the total measurement time, i.e., the number of photon-pairs consumed, and N_M for the shot number, the number of photon-pairs per unitary.

Discussion

In this work, we advance the RM framework by integrating BM and a few-shot post-processing strategies. We demonstrate its effectiveness in estimating PT moments for challenging mixed-state entanglement quantification. The BM-enhanced FSRM exhibits robustness to imperfections and noise in random unitary realization, while its few-shot feature ensures efficient post-processing. In addition, we introduce an error mitigation technique for BM to enhance the practicality of the scheme even further. It is worth mentioning that for an n-qubit state, only the BM between two qubits is necessary, ensuring scalability for large-scale quantum systems. Additionally, we find that, under a fixed measurement resource, the method with more versatile measurement settings with fewer repeats per setting is more efficient, challenging the previous estimation protocol with fixed number measurement settings and many repeats per setting.

We expect that BM-enhanced FSRM can be further applied to other important scenarios in quantum information and physics, like quantum correlation characterization^41,42,43, quantum error mitigation^44,45,46,47, and quantum algorithm design^48,49,50. Our research significantly enriches the scope of RM and advances it in real-world quantum experiments, especially in the characterization of photonic systems.

Methods

From RM to FSRM

Using the RM method to estimate properties of a quantum state is given by Eq. (4). Formally, to estimate a property ${{\mathcal{T}}}_{k}=Tr({T}_{k}{\rho }^{\otimes k})$, one must identify a function O_k that satisfies Eq.(3), i.e. ${{\mathbb{E}}}_{U}{U}^{\otimes k}{O}_{k}{U}^{\dagger \otimes k}={T}_{k}$. Here, we will show that, given the existence of such an O_k, all RM methods could be extended to the few-shot scenario.

There is a key observation that, with repetitive experiments, the expectation of any function of shot names satisfies:

$${{\mathbb{E}}}_{\{{\bf{b}}\}}{O}_{k}({{\bf{b}}}_{i,1},{{\bf{b}}}_{i,2},..,{{\bf{b}}}_{i,k})=\mathop{\sum }\limits_{\{{\bf{b}}\}\in {{\mathbb{F}}}_{2}^{nk}}{O}_{k}(\{{\bf{b}}\})P({{\bf{b}}}_{1})P({{\bf{b}}}_{2})...P({{\bf{b}}}_{k})$$

(10)

Here, ${{\mathbb{E}}}_{\{{\bf{b}}\}}$ is the expectation over repetitive experiments, and b_i,1,...,b_i,k in the left-hand side of the equation are the shot outcomes at the i-th round measurement with random unitary U_i. The right-hand of the equation means transverse all k-bases pairs {b}.

It should be noted that such repetitive experiments are virtual, which means we don’t really need to do the repetitive experiments, but a few-shot outcomes are enough. At this time, by combining Eqs. (4) and (10), the higher order covariance of the probabilities occurs naturally from the experiments without any post-processing and with merely a few shot outcomes enough. Say:

$${{\mathcal{T}}}_{k}(\rho )={{\mathbb{E}}}_{U}{{\mathbb{E}}}_{\{{\bf{b}}\}}\left[{O}_{k}({{\bf{b}}}_{i,1},{{\bf{b}}}_{i,2},..,{{\bf{b}}}_{i,k})\right]={\mathbb{E}}\left[{O}_{k}({{\bf{b}}}_{i,1},{{\bf{b}}}_{i,2},..,{{\bf{b}}}_{i,k})\right]$$

(11)

Here, ${\mathbb{E}}$ is the expectation of experiments by randomly selecting unitaries and collecting k shot outcomes from the experiment for each setting.

Equations of estimating p2 and p3 with FSRM

In this section, we discuss the estimation of the second and third moments of mixed-state entanglement using FSRM. For the entanglement quantification of unknown mixed-state, an important criterion called p₃-PPT, i.e., ${{\mathcal{N}}}_{3}={p}_{2}^{2}-{p}_{3}$¹⁹, was proposed, with ${p}_{k}=Tr[{({\rho }_{AB}^{{T}_{B}})}^{k}]$ the k-th moment of the PT density matrix^17,18. Importantly, such moments can be rewritten in the form ${p}_{k}=Tr[{W}_{\to }^{A}\otimes {W}_{\leftarrow }^{B}{\rho }^{\otimes k}]$, with W_←/→ is (clock/anti-clockwise) circular permutation operator on k-copy Hilbert space¹⁷.

We focus on an n-qubit system, with subsystems A and B containing n_A and n_B = n − n_A qubits. The second moment p₂ shows ${p}_{2}=Tr[{W}_{(12)}{({\rho }_{AB}^{{T}_{B}})}^{\otimes 2}]=Tr[{W}_{(12)}^{{T}_{B}}{\rho }_{AB}^{\otimes 2}]=Tr[{W}_{(12)}{\rho }_{AB}^{\otimes 2}]=Tr[{\rho }_{AB}^{2}]$, with W₍₁₂₎ the swap operator on 2-copy. Because ${W}_{(12)}={W}_{(12)}^{A}\otimes {W}_{(12)}^{B}$ with ${W}_{(12)}^{A/B}$ as the swap operator on the subspace A/B, we have ${W}_{(12)}^{{T}_{B}}={W}_{(12)}^{A}\otimes {W}_{(21)}^{B}={W}_{(12)}^{A}\otimes {W}_{(12)}^{B}={W}_{(12)}$. It is clear that p₂ also equals the purity of ρ and is thus independent of the partition.

Elben et al. proposed that with local random unitaries and computational-basis measurement, the purity can be expressed as follows^24,29:

$${p}_{2}={\text{Tr}}\left[{W}_{(12)}{\rho }^{\otimes 2}\right]=\mathop{\sum }\limits_{{{\bf{b}}}_{i,1},{{\bf{b}}}_{i,2}}d{(-2)}^{h({{\bf{b}}}_{i,1},{{\bf{b}}}_{i,2})}{{\mathbb{E}}}_{U}\left[P({{\bf{b}}}_{i,1}| {U}_{i})P({{\bf{b}}}_{i,2}| {U}_{i})\right],$$

(12)

where b_i,1 and b_i,2 are a n-bit shot outcomes collected in the i-th measurement round with untiary U_i, and h(b_i,1, b_i,2) is the humming distance between these two strings. Consequently, for the FSRM, the estimator of p₂ is directly

$${p}_{2}\to {O}_{2}({{\bf{b}}}_{i,1},{{\bf{b}}}_{i,2})={2}^{n}{(-2)}^{-h({{\bf{b}}}_{i,1},{{\bf{b}}}_{i,2})},$$

(13)

where b_i,1, b_i,2 are two sequential measurement outcomes under the same local random unitary ${U}_{i}{=\bigotimes }_{j=1}^{n}{u}_{i}^{(j)}$ with ${u}_{i}^{(j)}$ as local random unitary on the j-th qubit at i-th measurement round.

Actually, the post-processing function can be factorized into the product of single-qubit ones. For the 1-qubit case, it reads

$$n=1:\,{O}_{2}({{\bf{b}}}_{i,1},{{\bf{b}}}_{i,2})=\left\{\begin{array}{ll}2, & \,{{\bf{b}}}_{i,1}={{\bf{b}}}_{i,2};\\ -1, & \,{{\bf{b}}}_{i,1}\,\ne\, {{\bf{b}}}_{i,2}.\end{array}\right.$$

(14)

Next, we discussed the third moment p₃. The third moment reads ${p}_{3}=Tr[{({\rho }^{{T}_{A}})}^{3}]=Tr\left[({W}_{\to }^{A}\otimes {W}_{\leftarrow }^{B}){\rho }^{\otimes 3}\right]=Tr\left[({W}_{\leftarrow }^{A}\otimes {W}_{\to }^{B}){\rho }^{\otimes 3}\right]$, where we omit the subscript of ρ_AB for simplicity, and W_→ and W_← are cyclic permutation operators on the 3-copy Hilbert space. To be specific, the permutation operator ${W}_{\to }\left|{\psi }_{1}\right\rangle \left|{\psi }_{2}\right\rangle \left|{\psi }_{3}\right\rangle =\left|{\psi }_{3}\right\rangle \left|{\psi }_{1}\right\rangle \left|{\psi }_{2}\right\rangle$ and ${W}_{\leftarrow }\left|{\psi }_{1}\right\rangle \left|{\psi }_{2}\right\rangle \left|{\psi }_{3}\right\rangle =\left|{\psi }_{2}\right\rangle \left|{\psi }_{3}\right\rangle \left|{\psi }_{1}\right\rangle$. By defining ${M}_{neg}:=\frac{1}{2}({W}_{\to }^{A}\otimes {W}_{\leftarrow }^{B}+{W}_{\leftarrow }^{A}\otimes {W}_{\to }^{B})$, one further has${p}_{3}=Tr[{({\rho }^{{T}_{A}})}^{3}]=Tr[{M}_{neg}{\rho }^{\otimes 3}]$.

By further defining M₀ = W_→ + W_← and M₁ = W_→ − W_←. Zhou et al. put forward that it can be expressed as

$${M}_{neg}=\frac{1}{4}{M}_{0}^{A}\otimes {M}_{0}^{B}-\frac{1}{4}{M}_{1}^{A}\otimes {M}_{1}^{B},$$

(15)

in which the first part can be estimated from the local random unitaries and computational-basis measurement, and the second part can be estimated from local random unitaries and Bell-basis measurement¹⁸. We use ‘s’ to denote the measurement setting type, s = 0 for computational-basis measurement (CM) and s = 1 for Bell-basis measurement (BM).

To estimate M₀, only CM in the FRSM is required. First, considering a single-partite d-dimensional state $\rho \in {{\mathcal{H}}}_{d}$, the measurement result b_i = [0, 1 ⋯ d − 1]. By ref. ¹⁸ the estimation formula shows

$${\text{Tr}}[{M}_{0}{\rho }^{\otimes 3}]:\to {O}_{+}({{\bf{b}}}_{i,1},{{\bf{b}}}_{i,2},{{\bf{b}}}_{i,3}| s=0)=\left\{\begin{array}{cc}{d}^{2}+1, & \mathrm{wt}=3\\ 1-d, & \mathrm{wt}=2\\ 2, & \mathrm{wt}=1\\ & \,\end{array}\right.,$$

(16)

where wt(b_i,1, b_i,2, b_i,3) counts the same number among the inputs b_i,1, b_i,2, b_i,3. For a single-qubit system, i.e., d = 2, the estimation formula becomes

$$n=1:\,O({{\bf{b}}}_{i,1},{{\bf{b}}}_{i,2},{{\bf{b}}}_{i,3}| s=0)=1+{(-2)}^{{\rm{wt}}({{\bf{b}}}_{i,1},{{\bf{b}}}_{i,2},{{\bf{b}}}_{i,3})-1}=\left\{\begin{array}{ll}5, & {\rm{wt}}=3,\\ -1, & {\rm{wt}}=2.\end{array}\right.$$

(17)

Note that as the shot outcome b_i is a 1-bit number, it is impossible to have the wt = 0 case. Moreover, for a bipartite system, ${\rho }_{AB}\in {{\mathcal{H}}}_{{d}_{A}}\otimes {{\mathcal{H}}}_{{d}_{B}}$, the measurement result is divided into two parts, b = b^Ab^B, for subsystem A and B, and one has

$$Tr[({M}_{0}^{A}\otimes {M}_{0}^{B}){\rho }_{AB}]\to O({{\bf{b}}}_{i,1},{{\bf{b}}}_{i,2},{{\bf{b}}}_{i,3}| s=0)=O({{\bf{b}}}_{i,1}^{A},{{\bf{b}}}_{i,2}^{A},{{\bf{b}}}_{i,3}^{A}| s=0)O({{\bf{b}}}_{i,1}^{B},{{\bf{b}}}_{i,2}^{B},{{\bf{b}}}_{i,3}^{B}| s=0).$$

(18)

On the other hand, to estimate $Tr[({M}_{1}^{A}\otimes {M}_{1}^{B}){\rho }_{AB}^{\otimes 3}]$ for ${\rho }_{AB}\in {{\mathcal{H}}}_{{d}_{A}}\otimes {{\mathcal{H}}}_{{d}_{B}}$ with d_A = d_B = d, Zhou et al. proved that BM is required across A and B after the local random unitary evolution. We denote the three shot outcomes for BM result as β_i,1, β_i,2 and β_i,3, represented by a d²-bit string β_i = [0, 1,⋯ , d² − 1]. To be specific, Eq. (52) in the Supplementary Material of ref. ¹⁸ provides the post-processing functions as follows. $O({\rm{wt}}=3)={d}^{2}{({d}^{2}-1)}^{2}$, $O({\rm{wt}}=2)=-3{d}^{2}{({d}^{2}-1)}^{2}$, and $O({\rm{wt}}=1)=\frac{1}{2}{d}^{4}{({d}^{2}-1)}^{2}$ or $-\frac{1}{2}{d}^{2}{({d}^{2}-1)}^{2}({d}^{2}-4)$, where wt(β_i,1, β_i,2, β_i,3) still represents the number of the same elements. The coefficients provided by ref. ¹⁸ include the number of occurrences of different bases. By considering the number of all kinds of scenarios, #(wt = 3) = d, #(wt = 2) = d(d − 1), and #(wt = 1) = d(d − 1)(d − 2). The estimation formula used in our experiments is thus as follows

$${\text{Tr}}[({M}_{1}^{A}\otimes {M}_{1}^{B}){\rho }^{\otimes 3}]\to O({\beta }_{i,1},{\beta }_{i,2},{\beta }_{i,3}| s=1)=\left\{\begin{array}{cc}{({d}^{2}-1)}^{2} & \mathrm{wt}=3,\\ (1-{d}^{2}) & \mathrm{wt}=2,\\ {d}^{2}({d}^{2}-1)/2({d}^{2}-2) & \mathrm{wt}={1}^{* }.\end{array}\right.$$

(19)

For wt = 1, the cases are complicated, and only some wt = 1 cases are applicable for this expression, with further details being found in ref. ¹⁸. However, for a two-qubit system, i.e., a 2 × 2 system, the estimation formula can be simplified to be

$$n=2:\,O({\beta }_{i,1},{\beta }_{i,2},{\beta }_{i,3}| s=1)=1-{(-2)}^{{\rm{wt}}({\beta }_{i,1},{\beta }_{i,2},{\beta }_{i,3})}=\left\{\begin{array}{ll}9 & {\rm{wt}}=3,\\ -3 & {\rm{wt}}=2,\\ 3 & {\rm{wt}}=1.\end{array}\right.$$

(20)

where β_i,1/2/3 ∈ [0, 1, 2, 3], and the formula depends on the value of wt.

In a 2-qubit system, from Eqs. (18) and (20), it is clear that both b and β can be labeled by 2-bit, and in this way we can denote them with a single label as c_i,1, c_i,2, and c_i,3 from the set 00, 01, 10, 11. Therefore, to estimate p₃ of a 2 × 2 system by implementing CBM and BSM with a 50–50% distribution, one has

$${p}_{3}={\text{Tr}}[{M}_{neg}{\rho }_{AB}^{\otimes 3}]\to \frac{{(-1)}^{s}}{2}O({{\bf{c}}}_{i,2},{{\bf{c}}}_{i,2},{{\bf{c}}}_{i,3}| s).$$

(21)

Finally, Eq. (21) can be extended to an n-qubit system by randomly implementing BMs between qubit pairs. We use an n-bit string $\overrightarrow{a}$ to denote the partition, where ’0’ represents subsystem A, and ’1’ for B. We consider another n-bit string $\overrightarrow{s}$ to denote the measurement setting with an even number of ’1’s. If two bits are both ’1’, we implement a BM between the corresponding qubits. The outcome of the BM on the $(j,j^{\prime} )$ qubit-pair is denoted as ${\beta }^{(j,j^{\prime} )}=0,1,2,3$. Note that for a given $\overrightarrow{s}$, it may correspond to a few different measurement settings, and we just fix one setting. For instance $\overrightarrow{s}=1111$ of 4-qubit system, say BM on qubit-pairs (12)(34) or (14)(23).

The operator in Eq. (15) can be further written into

$${M}_{neg}=\frac{1}{{2}^{n}}\mathop{\sum }\limits_{\overrightarrow{s},\wp (\overrightarrow{s})=0}{(-1)}^{(\overrightarrow{a}\cdot \overrightarrow{s})}\underset{k=1}{\overset{n}{\bigotimes }}{M}_{{s}_{k}},$$

(22)

where $\wp (\overrightarrow{s})=0$ represents the parity of $\overrightarrow{s}$ being zero, i.e., it contains an even number of “1"s, and $\overrightarrow{a}\cdot \overrightarrow{s}$ denotes the dot product between $\overrightarrow{a}$ and $\overrightarrow{s}$. Here, s_k represents the kth bit of the string $\overrightarrow{s}$.

To estimate p₃ = Tr[M_negρ^⊗3], we evolve the quantum state with local random unitary $U{=\bigotimes }_{j=1}^{n}{u}^{(j)}$ and choose the final measurement based on $\overrightarrow{s}$. We uniformly sample $\overrightarrow{s}$ with $\wp (\overrightarrow{s})=0$ and perform corresponding measurements. There are 2ⁿ⁻¹ of $\overrightarrow{s}$ that have an even number of ’1’s. Therefore, the probability is 1/2ⁿ⁻¹. We need to collect three sequential shot outcomes from the experiments to return a basic estimation. Note that, in each shot, the random unitary U and the measurement setting $\overrightarrow{s}$ should be kept the same.

The final estimator of p₃ reads ${O}_{3}({{\bf{b}}}_{1},{{\bf{b}}}_{2},{{\bf{b}}}_{3})={\sum }_{\overrightarrow{s}}\,\Pr \,(\overrightarrow{s}){O}_{3}({{\bf{b}}}_{1},{{\bf{b}}}_{2},{{\bf{b}}}_{3}| \overrightarrow{s})$ with ${\rm{g}}(\mu )={f}_{1}({{\rm{x}}}_{1})+{f}_{2}({{\rm{x}}}_{2})+\cdots +{f}_{{\rm{n}}}({{\rm{x}}}_{{\rm{n}}})\,+\varepsilon$ with

$${O}_{3}({{\bf{b}}}_{i,1},{{\bf{b}}}_{i,2},{{\bf{b}}}_{i,3}| \overrightarrow{s})=\frac{1}{2}{(-1)}^{\overrightarrow{a}\cdot \overrightarrow{s}}\mathop{\prod }\limits_{j:{s}_{j}=1}f(j,j{\prime} )\mathop{\prod }\limits_{j:{s}_{j}=0}g(j).$$

(23)

Here f and g functions are for the BM and CM outcome of qubit-pair $(j,j{\prime} )$ and qubit j, determined by the vector $\overrightarrow{s}$, and they show

$$\left\{\begin{array}{l}f(j,j^{\prime} )=1-{(-2)}^{{\rm{wt}}({\beta }_{i,1}^{(j,j{\prime} )},{\beta }_{i,2}^{(j,j{\prime} )},{\beta }_{i,3}^{(j,j{\prime} )})}\\ g(j)=1+{(-2)}^{[{\rm{wt}}({{\bf{b}}}_{i,1}^{(j)},{{\bf{b}}}_{i,2}^{(j)},{{\bf{b}}}_{i,3}^{(j)})-1]}\end{array}\right.,$$

(24)

where wt() counts the number of same elements for β and b, respectively. In the case of a 2-qubit system, with A and B containing one qubit each, i.e., $\overrightarrow{a}=10$. The vector $\overrightarrow{s}=00,11$, which indicates that one conducts CM and BM with equal probability on the 2-qubit system. By Eq. (7), the post-processing is reads

$${O}_{3}=\left\{\begin{array}{l}\frac{1}{2}g(1)g(2),\,\overrightarrow{s}=00,\\ -\frac{1}{2}f(1,2),\,\overrightarrow{s}=11.\end{array}\right.$$

(25)

which is consistent with Eq. (21).

CS and FSRM methods under noise

In this part, we discuss the robustness of both the classical Shadow (CS) and Randomized Measurement (RM) schemes, using a simple Gaussian noise model as a case study. It shows that the RM and FSRM schemes are more robust than the CS scheme, considering the noise in the random unitary.

The CS scheme and FSRM scheme both use the projective measurement result from the random unitary evolution on the sequentially prepared unknown quantum state, but they mainly diverge in the post-processing of these data. Mathematically, the effective quantum channel involved in the post-processing is quite different. For the CS scheme, the channel is a measurement and preparation channel in the form

$${\mathcal{M}}(\rho )={{\mathbb{E}}}_{U}\mathop{\sum }\limits_{{\bf{b}}}\Pr ({\bf{b}}| U){U}^{\dagger }\left|{\bf{b}}\right\rangle \left\langle {\bf{b}}\right|U,$$

(26)

with the (conditional) probability $\Pr ({\bf{b}}| U)=\left\langle {\bf{b}}\right|U\rho {U}^{\dagger }\left|{\bf{b}}\right\rangle$ by Born’s rule. Suppose the measurement is tomographic (over)complete, the inverse of the channel ${\mathcal{M}}$ then exists, and thus the unbiased estimation of ρ then returns as $\widehat{\rho }={{\mathcal{M}}}^{-1}(\left|{\bf{b}}\right\rangle \left\langle {\bf{b}}\right|)$. The CS scheme serves as a universal estimator for linear and nonlinear tasks. In particular, for nonlinear functions like ${{\mathcal{T}}}_{k}$, one can construct the estimator $Tr[{T}_{k}{\widehat{\rho }}^{({i}_{1})}\otimes \cdots \otimes {\widehat{\rho }}^{({i}_{k})}]$ from k distinct classical snapshots. On the other hand, as mentioned in Supplementary Note 1, the RM scheme specializes in some nonlinear properties, like purity. The effective channel is the k-fold twirling channel as ${\Phi }^{(k)}(\rho )={{\mathbb{E}}}_{U}[{U}^{\otimes k}{\rho }^{\otimes k}{U}^{\dagger \otimes k}]$. In particular, the FSRM scheme is good at predicting properties that are invariant under this twirling channel.

These two channels react differently when faced with errors or noise introduced alongside the random unitaries. For example, ref. ⁵¹ discusses CS in the context of various types of noise, such as the depolarizing channel and phase-damping channel. In these cases, the conditional probability Pr(b∣U) in CS changes

$$\langle {\bf{b}}| U\rho {U}^{\dagger }| {\bf{b}}\rangle \to \langle {\bf{b}}| \xi (U\rho {U}^{\dagger })| {\bf{b}}\rangle ,$$

(27)

with ξ for the noise channel. If the experimenter is unaware of this change, the classical preparation remains unchanged. The overall measurement and preparation channel is changed to

$$\widetilde{{\mathcal{M}}}(\rho )={{\mathbb{E}}}_{U}{\sum }_{{\bf{b}}}\langle {\bf{b}}| \xi (U\rho {U}^{\dagger })| {\bf{b}}\rangle \,{U}^{\dagger }\left|{\bf{b}}\right\rangle \left\langle {\bf{b}}\right|U,$$

(28)

and there is a mismatch between the measurement and preparation. As a result, if one keeps the original inverse ${{\mathcal{M}}}^{-1}$, the estimator ${{\mathcal{M}}}^{-1}({U}^{\dagger }\left|{\bf{b}}\right\rangle \left\langle {\bf{b}}\right|U)$ should be biased, and introduces systemic error in the estimation which makes CS not robust. Note that ref. ⁵² tactfully introduces a unified method to calibrate and mitigate such kinds of noises. In particular, it identifies the updated channel $\widetilde{{\mathcal{M}}}$, with the assumptions that the state $\left|{\bf{0}}\right\rangle$ is perfect and the noisy channel ξ is gate-independent and Markovian.

In summary, such an error issue in CS lies in the fact that one needs to have the exact information of the executed random unitary U. On the other hand, in the RM scheme, the k-fold twirling channel Φ^(k) can suffer some noise by itself since one does not need to know the exact information of each realization of U, but just keeps the unitary random enough. In particular, it is robust to some coherent errors.

Experimental setup

The experimental setup is illustrated in Fig. 6A. The left part is the polarization-entangled state generation setup, which is the same as the setup in ref. ³⁵. A continuous-wave laser operating at 404 nm generates photons with initial polarization state ρ₀. These violet photons are coupled into a Sagnac interferometer where horizontally polarized ($\left|H\right\rangle$) photons propagate clockwise while vertically polarized ($\left|V\right\rangle$) photons propagate counterclockwise. A dichroic half-waveplate (Di-HWP) converts the counter-propagating $\left|V\right\rangle$ violet photons to $\left|H\right\rangle$ polarization before they pump a periodically poled potassium titanyl phosphate (PPKTP) crystal from both directions. Through parametric down-conversion (PDC), each $\left|H\right\rangle$ pump photon generates $\left|H\right\rangle$ and $\left|V\right\rangle$ photon pairs at 808 nm. The Sagnac interferometer then separates these photons into distinct output ports based on their propagation directions, generating polarization-entangled photon pairs.

Fig. 6: Full setup and input states of our experiment.

A Full setup. Compared to Fig. 3, the left part of the schematic has been added to illustrate the generation of polarization-entangled photon pairs. A continuous-wave laser with a central wavelength of 404 nm pumps a Sagnac-type interferometer containing a periodically poled potassium titanyl phosphate (PPKTP) crystal. Inside the crystal, a type-II spontaneous parametric down-conversion (SPDC) process generates correlated photon pairs at 808 nm with orthogonal polarizations (H and V). The Sagnac interference, combined with a dichroic half-waveplate (Di-HWP), a dichroic polarization beam splitter (Di-PBS), a low-pass filter (LPF), and a band-pass filter (BPF), ensures efficient separation and collection of the entangled photon pairs. By adjusting the polarization of the 404-nm laser, we could generate a maximum-entangled state and a partial-entangled state. B Tomography for the pure maximum-entangled pure state with the fidelity of 97.679%. C Tomography for the one mixed state with eigenvalues of 0, 0, 0.077, and 0.923.

When the initial violet photon is prepared in the $\frac{1}{\sqrt{2}}(\left|H\right\rangle +\left|V\right\rangle )$, the output state ideally forms a maximally entangled Bell state, $\frac{1}{2}(\left|HV\right\rangle +\left|VH\right\rangle )$. By carefully adjusting the initial pump polarization ρ₀ and introducing a mismatch of the clockwise and counter-clockwise paths to the Sagnac interferometer, we can introduce mixedness to produce various entangled mixed states.

To verify the state quality, we perform quantum state tomography on both the maximally entangled and mixed states using Pauli measurements implemented in the outer section of the setup (Fig. 6A). The tomographic reconstruction via maximum likelihood estimation (MLE) yields a fidelity of 97.68% for the Bell state (Fig. 6B), confirming successful generation of high-quality entanglement. For the mixed state (Fig. 6c), the reconstructed density matrix exhibits eigenvalues of 0, 0, 0.077, and 0.923, demonstrating preparation of a non-maximally entangled mixed state.

In theory, we intended to implement a complete Bell basis measurement. However, achieving this goal proved impossible without resorting to post-selection. Following the protocol⁵³, we opted to implement an optical CZ gate and some Hadamard gates ($\widehat{H}$) before a ZZ measurement to form a complete Bell measurement. That is, the gates $(I\otimes \widehat{H})CZ(\widehat{H}\otimes \widehat{H})$ could turn the four Bell state ($\frac{1}{\sqrt{2}}(\left|00\right\rangle \pm \left|11\right\rangle )$ and $\frac{1}{\sqrt{2}}(\left|01\right\rangle \pm \left|10\right\rangle )$) into the four $\left|00\right\rangle ,\left|01\right\rangle ,\left|10\right\rangle$ and $\left|11\right\rangle$. And the optical CZ gate in our experiment follows the setup in refs. ^37,38,39.

The composition of the CZ gate involves three partial-polarization-beam-splitters (PPBSs) with horizontal polarization transmission of t_H = 1 and vertical polarization transmission and reflection values of t_V = 1/3 and r_V = 2/3, respectively. That turns the ${a}_{V,1}^{\dagger }\to \sqrt{\frac{1}{3}}{a}_{V,1}^{\dagger }+i\sqrt{\frac{2}{3}}{a}_{V,2}^{\dagger }$ and ${a}_{V,2}^{\dagger }\to -i\sqrt{\frac{2}{3}}{a}_{V,1}^{\dagger }+\sqrt{\frac{1}{3}}{a}_{V,2}^{\dagger }$, and polarization-H photon unchanged. The two photons undergo interference on the first PPBS, with subsequent PPBS units on the two arms attenuating horizontal polarization to 1/3. In this setup, Hong-Ou-Mandel (HOM) interference occurs only when the input state is $\left|VV\right\rangle$. This leads to a reduction in coincidence from 5/9 to 1/9, resulting in a success rate of 1/9 for the optical CZ gate. The HOM interference between two vertically polarized photons is illustrated in Fig. 7.

Fig. 7: Hong-Ou-Mandel interference as a function of path difference.

The x-axis denotes the path difference between the two branches of the first PPBS, and the y-axis shows the corresponding photon counts. The blue dots and curve correspond to the input polarization state $\left|V\right\rangle \left|V\right\rangle$ at the first PPBS without other PPBS or HWPs. The orange dots and curve correspond to the input state $\left|V\right\rangle \left|+\right\rangle$ for the entire CNOT part (composed of two Hadamard gates and a CZ gate implemented with three PPBSs), with measurements performed in the $\left|V\right\rangle \left|+\right\rangle$ basis.

We use 1 and 2 as two sides of the PPBS. We could summarize how the three PPBS and HOM generate the CZ gate as follows:

$$\begin{array}{rcl} & & \left({c}_{HH}{a}_{H,1}^{\dagger }{a}_{H,2}^{\dagger }+{c}_{HV}{a}_{H,1}^{\dagger }{a}_{V,2}^{\dagger }+{c}_{VH}{a}_{V,1}^{\dagger }{a}_{H,2}^{\dagger }+{c}_{VV}{a}_{H,1}^{\dagger }{a}_{H,2}^{\dagger }\right)\left|0\right\rangle \\ \mathop{\longrightarrow }\limits^{1\mathrm{st}\,\mathrm{PPBS}\,{\& }\,\mathrm{HOM}} & & \left({c}_{HH}{a}_{H,1}^{\dagger }{a}_{H,2}^{\dagger }+\sqrt{\frac{1}{3}}{c}_{HV}{a}_{H,1}^{\dagger }{a}_{V,2}^{\dagger }+\sqrt{\frac{1}{3}}{c}_{VH}{a}_{V,1}^{\dagger }{a}_{H,2}^{\dagger }-\frac{1}{3}{c}_{VV}{a}_{H,1}^{\dagger }{a}_{H,2}^{\dagger }\right)\left|0\right\rangle \\ \mathop{\longrightarrow }\limits^{2\mathrm{nd}\,{\& }\,3\mathrm{rd}\,\mathrm{PPBS}} & & \frac{1}{3}\left({c}_{HH}{a}_{H,1}^{\dagger }{a}_{H,2}^{\dagger }+{c}_{HV}{a}_{H,1}^{\dagger }{a}_{V,2}^{\dagger }+{c}_{VH}{a}_{V,1}^{\dagger }{a}_{H,2}^{\dagger }-{c}_{VV}{a}_{H,1}^{\dagger }{a}_{H,2}^{\dagger }\right)\left|0\right\rangle \end{array}$$

(29)

More details see ref. ³⁸.

Unlike the CZ gate, the CNOT gate exhibits asymmetry in choosing the control and target arms. We designated one arm as the target arm by introducing two Hadamard gates (each with a rotation angle of 22. 5^∘) before and after the optical CZ gate. Proper application of the CNOT gate demonstrates that $\left|V\right\rangle \left|+\right\rangle \to CNOT\left|V\right\rangle \left|-\right\rangle$, where $\left|\pm \right\rangle =\frac{1}{\sqrt{2}}(\left|H\right\rangle \pm \left|V\right\rangle )$. This results in a pronounced dip in the HOM interference, as depicted in Fig. 7.

To enhance the HOM interference, which relies on wavelength correlation between the two photons, we employed two tiled 3nm filters’ edges to filter the wavelengths (not shown in Fig. 6).

In practical applications, the PPBSs exhibit imperfections, introducing a reflection coefficient of r_H ≈ 1/50 and causing changes in polarization. Importantly, these imperfections result in an error rate, with our target being attenuated to 1/9. This significant reduction in the signal-to-noise ratio poses a challenge, making it difficult to achieve the fidelity of the optical CNOT gate exceeding 80%. Consequently, we recognize the necessity for a measurement imperfection amendment in our experiments to accurately predict p₃.

Results with the maximum-entangled state

In the experiments, we produced a maximally entangled state to verify our protocols. The results of tomography are shown in Fig. 6b with a fidelity of 97.68%. For a maximally entangled state, p₂ = 1 and p₃ = 0.25. Results of our prediction from 10 independent experiments are depicted in Fig. 8. Utilizing the FSRM scheme and predicting from 10⁴ response numbers, we obtain experimental estimations of ${\widehat{p}}_{2}=0.9983\pm 0.0108$ and ${\widehat{p}}_{3}=0.2322\pm 0.0078$ (see Fig. 8a, c), converging to the true state values. Additionally, we estimate the mixed state entanglement as ${p}_{2}^{2}-{p}_{3}=0.7638\pm 0.0294$. The obtained entanglement value exceeds 0, affirming the existence of mixed-state entanglement. The uncertainties in the estimations stem from 10 independent experiments. However, employing the one-shot CS scheme using the same dataset as the FSRM scheme, we estimate ${\widehat{p}}_{2}=0.6454\pm 0.0112$ and ${\widehat{p}}_{3}=0.5185\pm 0.0253$ (see Fig. 8b, d). Our analysis shows that the discrepancy in predictions between the CS and FSRM schemes arises from the need for exact information on the unitaries in the CS scheme, leading to accumulated unitary errors. In contrast, for the FSRM scheme, no such information is necessary. This discrepancy aligns with our analysis regarding Gaussian noise in unitaries, as detailed in Supplementary Note 3. Consequently, FSRM proves to be more practical in noisy environments.

Fig. 8: Experimental prediction from a pure state.

We compare the prediction accuracy for estimating p₂ and p₃ between the FSRM scheme (A, C) and CS schemes (B, D). In the FSRM approach, we estimate p₂ and p₃ using 2 and 3 shot outcomes per unitary, respectively. Error mitigation techniques are applied to enhance the estimation of p₃. Each experiment is repeated independently ten times, denoted by different lines. For the CS scheme, only one shot per unitary is required for prediction. Additionally, to ensure a fair comparison with FSRM, we utilize the same dataset for CS by selecting one shot from the two or three collected shot outcomes per unitary, particularly if the measurement is on the computational bases. By comparing the results of (A–D), although both schemes utilize the same dataset, FSRM demonstrates convergence to the true values, while the CS scheme does not.

Figure 9 shows the scaling with N_M measurements per unitary and demonstrates similar results with the mixed state (Fig. 4 in the main text). Through linear regression, we ascertain $\log {\rm{Err}}=-0.5\log N+0.2979$ as N_M = 2, and for RM (N_M = 200), $\log {\rm{Err}}=-0.5\log N+0.7735$, which is labeled by a cross mark in Fig. 9. Consequently, utilizing FSRM results in a 4.75 dB improvement in estimation compared to the original RM.

Fig. 9: Statistic error performance in prediction of p₂ with FSRM for maximum-entangled state.

N for the total measurement time, i.e., the number of photon-pairs consumed, and N_M for the shot number, the number of photon-pairs per unitary.