Experimental demonstration of efficient quantum state tomography of matrix product states

Yuan-Yuan Zhao; Zhibo Hou; Guo-Yong Xiang; Yong-Jian Han; Chuan-Feng Li; Guang-Can Guo

doi:10.1364/OE.25.009010

1. Introduction

Quantum states are the foundation of quantum mechanics and quantum information. The structure of the Hilbert space causes the parameters of quantum states to increase exponentially with the system size. This exponential complexity enables implementation of quantum processes, such as quantum computation, that are more efficient than their classic counterparts. However, the complexity of many-body states also makes it difficult to determine their concrete form. In general, a quantum state (pure or mixed) can be fully determined using quantum state tomography [1, 2] to measure the complete set of observables. The quantum state can then be reconstructed by various methods such as maximum likelihood estimation (MLE), Bayesian mean estimation (BME) [3] and linear regression estimation (LRE) [4]. However, application of standard quantum state tomography to many-body states quickly becomes intractable because the required number of measurements and the post-processing time increase exponentially with the size of the system. Nevertheless, many-body quantum states inevitably appear in quantum information and technology, particularly in quantum computation, and the improvement of such technologies has enabled the preparation of quantum states with many qubits in several systems, such as ten-photon quantum states in optical systems [5] and 14-ion GHZ states in ion traps [6]. Therefore, there is an urgent need to develop efficient quantum state tomography methods for many-body quantum states.

The relationship between the global pure state and the local reduced-density matrices [7–9] will help efficient quantum state Tomography. Linden et al. [10] firstly noted that nearly every three-qubit pure state, except for certain special states (those that form a set with whose volume is zero in the Hilbert space), can be fully determined from its two-qubit reduced-density matrices. This result was recently demonstrated experimentally in [11]. Linden et al. [12] further noted that nearly every N-qubit pure state can be completely determined by its reduced-density matrices and that the required size of these reduced matrices can be less than 2N/3 [13]. The fact that a global pure state can be completely determined by its reduced-density matrices can be used to dramatically reduce the number of local measurements required to reconstruct the global state. Unfortunately, Linden et al. [10, 12] did not develop an algorithm to reconstruct the global pure state from its reduced-density matrices. However, recent mathematical developments have yielded a method, known as singular value thresholding (SVT) algorithm [14], which originally propose to approximate the matrix with minimum nuclear norm obeying a set of convex constraints. With this algorithm, when a matrix is low-rank or can be well approximated by a low-rank matrix, this matrix can be recovered from a sampling of its entries. In our situation, for the pure (or near pure) state, the significant rank will be very low (rank 1 for pure state), the algorithm can be used to reconstruct a global density matrix from its local reduced-density matrix [15–17]. But the time required to reconstruct a general global state still scales exponentially with the number of particles for straightforward application of the SVT method.

However, if we focus on the one-dimensional matrix product states (1D MPSs) [18], which are pure states and can be accurately determined using polynomial scale parameters (see Eq. 1), the entire reconstruction process, including the measurements and post-processing, can be completed efficiently [19]. The 1D MPSs are general in 1D quantum system: actually, any 1D quantum pure states can be expressed as an MPS using a suitable value for the Schmidt number D, which may be very large in some cases. It has been theoretically proven that the ground state of a 1D gapped system can be reliably approximated by an MPS with a relatively small D value [20, 21]. More importantly, many interesting quantum states in quantum information and quantum computation, such as W states and 1D Affleck-Kennedy-Leib-Tasaki states [22], can be described as an MPS with a very small D value. Cramer et al. showed in a theoretical study [19] that an MPS can be reconstructed from its reduced-density matrices with saturated sizes which can be estimated as ⌊log D⌋ +1. The number of reduced-density matrices that are required to reconstruct the global pure state scales linearly with the size of the system. Furthermore, an efficient algorithm has been developed that is based on a modified SVT algorithm and tensor network algorithm [23] and can be used to reconstruct the many-body state from the reduced-density matrices with a polynomial resource. In this work, we are aimed to experimentally demonstrate the validity of the quantum state tomography introduced in [19], and show that the method is robust for noise.

2. Matrix product states

The matrix product states can be expressed in the following form [18] for an open-boundary condition:

| Ψ_{MPS} 〉 = \sum_{s_{1} \dots s_{N} = 1}^{d} Tr (A_{1}^{s_{1}} A_{2}^{s_{2}} \dots A_{N}^{s_{N}}) | s_{1} \dots s_{N} 〉,

where d is the dimension of the physical indices s_k (d = 2 for a qubit). For a fixed physical index s_k,

A_{k}^{s_{k}}

is a D × D matrix at site k, where D is the Schmidt number that indicates the entanglement of this state, and D is determined by the rank of the corresponding reduced density matrix. Therefore, the slightly entangled 1D state [18] can be expressed as an MPS with a very small D value. For example, the state,

| Ψ^{4} 〉 = \frac{\sqrt{3}}{3} (| H H H H 〉 + | V V V V 〉) + \frac{\sqrt{3}}{6} (| H H V V 〉 + | H V V H 〉 + | V H H V 〉 + | V V H H 〉)

(details in the experiment part Eq. 4) is an MPS D = 3, which can be directly determined by the rank of the reduced-density matrix tr₁₂(|Ψ⁴〉〈Ψ⁴|).

3. Algorithm for reconstructing an MPS from its reduced density matrices

According to the algorithm in [19], we can obtain the reduced-density matrices with very few qubits by standard quantum state tomography for a n-qubit MPS. Let $P_{m}^{i}$ be equal to the possible products of the Pauli operators that act on sites i, i + 1, …, i + k and is trivial on the other qubits (k is the number of qubits in the reduced-density matrices that can be roughly approximated by ⌊log D⌋ + 1). Then, a k-qubit reduced-density matrix can be expanded as $ρ_{i} = \frac{1}{2^{N}} \sum_{m} t r [ρ_{i} P_{m}^{i}] P_{m}^{i}$ , where the expectations, $t r [ρ_{i} P_{m}^{i}]$ , can be directly measured in experiments. The modified SVT algorithm can reconstruct the global MPS efficiently from these reduced-density matrices. The algorithm can be described as:

\begin{array}{l} {\hat{X}}_{n} = - y_{n} \sum_{m . i} \frac{〈 y_{n} | P_{m}^{i} | y_{n} 〉}{2^{N}} P_{m}^{i} \\ {\hat{Y}}_{n + 1} = {\hat{Y}}_{n} + δ_{n} (\hat{R} - {\hat{X}}_{n}) . \end{array}

where the operator

\hat{R} = \frac{1}{2^{N}} \sum_{m, i} t r [ρ_{i} P_{m}^{i}] P_{m}^{i}

is constructed, and the initial matrix

{\hat{Y}}_{0}

is taken as the zero matrix. The parameter δ_n is chosen to guarantee the convergence. The final global n-qubit pure state can be recursively estimated by finding the eigenstate |y_n〉 that corresponds to the largest eigenvalue of

{\hat{Y}}_{n}

. Unfortunately, the resource used to directly find the eigenvector |y_n〉 still scales exponentially with the system size. As stated in [19], the locality of

P_{m}^{i}

and the methods used in condensed-matter studies [23] (such as, imaginary time evolution algorithm based on tensor network states [20]) can be used to efficiently find the eigenstate with the largest eigenvalue as the ground state of a Hamiltonian with a local interaction. Therefore, the entire process, including measurement and post-processing, for reconstructing a pure many-body quantum state scales as a polynomial function in the number of qubits.

However, in practice, two important issues should be further considered. First, whereas an MPS is a pure state in theory [19], the prepared quantum state is inevitably affected by the environment and is therefore a mixed state with some fidelity. As mentioned in [19], a direct application of the algorithm developed by Cramer et al. to reconstruct a mixed state from its reduced-density matrices yields a result that is not unique, and the fidelity between the reconstructed mixed state and the prepared state increases with the size of the reduced matrices. In other words, the saturation of the size of the reduced density matrices in this reconstruction method, which is the key requirement for the efficient state tomography, fails in the mixed state reconstruction. Actually, some kind of many-body mixed states (thermal state of a gapped system as an example) can be efficiently reconstructed by the maximum likelihood method [33]. In this method, the MPS can be extended to matrix product operators with low-bond dimension. And a scalable maximum likelihood algorithm, which relies on a well-established fixed point algorithm for maximizing the likelihood function, can be used to reconstruction the global quantum state. Second, measurement errors in the reduced-density matrices is unavoidable. In addition, statistical errors are inevitable during the reconstruction of the local-reduced density matrices. If a set of reduced-density matrices originates from a global pure state, they must satisfy certain compatibility conditions [24, 25]. (Necessary and sufficient conditions have been found for one-qubit reduced-density matrices that originate from a pure state [26]. However, for the more important two-qubit or the three-qubit case [27], these conditions remain an open question.) Reduced-density matrices with measurement and statistical errors can potentially violate the compatibility conditions such that they can not obtained from a common pure quantum state. Thus, reconstructing the global quantum state from the local reduced-density matrices with errors are highly nontrivial.

To demonstrate the validity of a theory for a pure state and study the effect of errors, we suppose that we have no information about the errors or noises of the global state and the reduced density matrices, and the obtained reduced density matrices are come from a pure state and we still use the algorithm introduced in [19] to reconstruct the pure state. If there is no noise and error, the reconstruction is valid according to [19], however, for the noisy prepared state and noisy local reduced-density matrices, the validity and the robustness of the reconstruction algorithm is questionable. For a mixed global state, we need to modify the rank of the reconstructed state to 1 in the algorithm of [14] and the convergence of the method can be obtained in the same way. Actually, the density matrices of quantum states form a convex set and the pure states are located at the extreme points. From geometric point of view, the algorithm is to find the nearest extreme points which are supposed to be very robust against noise. As demonstrated in our experiment with mixed states with different fidelities,the algorithm is indeed robust.

4. Experimental results

In our experiment, a type-II BBO crystal is pumped by a 390-nm femtosecond laser pulse, which is generated by frequency doubling the output of a mode-locked Ti:sapphire laser (with a 76-MHz repetition rate and a 780-nm wavelength) (Fig. 1). This SPDC process and the half-wave plate before BS1 produces a four-photon state:

| Ψ 〉 = {(a_{H}^{†} a_{H}^{†} + a_{V}^{†} a_{V}^{†})}^{2} | 0 〉 .

After BS1 and BS2, the aforementioned state is transformed into the ideal state [28]

\begin{array}{l} | Ψ^{4} 〉 = \frac{\sqrt{3}}{3} (| H H H H 〉 + | V V V V 〉) \\ + \frac{\sqrt{3}}{6} (| H H V V 〉 + | H V V H 〉 + | V H H V 〉 + | V V H H 〉) . \end{array}

Fig. 1 Experimental configuration: an ultraviolet (UV) pulse pumps a 2 mm beta-barium-borate (BBO) crystal to produce two pairs of entangled photons; beam splitter 1 (BS1) and BS2 each split one pair of photons into two arms; four sets of quarter-wave plates (λ/4), half-wave plates (λ/2) and polarizing beam splitters (PBS) are used to perform quantum state tomography; the birefringence of o and e in the BBO is compensated by BBO1 and BBO2, and the phase difference between the H and V polarizations in the reflected path after BS1 (BS2) is compensated by Q1 (Q2) generated by tilting the quarter-wave plate to the left or the right; quartz plates with different lengths are used as dephasing channels

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It should be noted that the state demonstrated here is permutationally invariant. For this kind of states, an efficient tomography method has been experimentally demonstrated in [29], both the number of measurement basis and the post-processing time scale polynomially with the number of qubits. Here we use the state to demonstrate the validity and robustness of the method introduced in [19] due to the experimental convenience.

In Fig. 1, BBO1 and BBO2 are used to compensate the birefringence of o (ordinary ray) and e (extraordinary ray) in BBO; however, in practice, this birefringence cannot be completely compensated. To perform the full quantum state tomography in a reasonable amount of time, we use 500 mw of pump light to obtain a high rate of four photon counts (this laser power yields approximately 70 four-fold coincidence counts per minute), which inevitably introduces the high-order-term noise. Therefore, the prepared states are not perfect and have a fidelity of 0.9422 ± 0.0041. However, to the best of our knowledge, this result is the highest reported fidelity for a four-photon entangled state [30]. We use the standard quantum state tomography to get the prepared states and the reduced-density matrices. We obtain the two-photon density matrices, ϱ₁₂, ϱ₂₃, and ϱ₃₄, when the other two photons are triggered, and the three-photon density matrices, ϱ₁₂₃ and ϱ₂₃₄, when the fourth photon is triggered. We use the two- or three-photon density matrices obtained by quantum tomography with measurement and statistical errors, and the algorithm in [19] (the parameter δ_n = 0.002 for two-photon case and δ_n = 0.6 for three-photon case) to obtain a pure state. Fig. 2 shows the real part of the densities of the ideal pure state, the prepared state obtained by standard tomography, the pure state reconstructed from the two-photon reduced density matrices (MPS2), and the pure state reconstructed from the three-photon reduced density matrices (MPS3). For the state we prepared, the results clearly show that the reconstructed pure states from the two-photon reduced matrices (|ϕ₂〉) are almost the same as those from the three-photon reduced matrices |ϕ₃〉 (the fidelity between |ϕ₂〉 and |ϕ₃〉 is 0 9934 ± 0.0022), and that these reconstructed pure states are very close to the ideal pure state (Fig. 2).

Fig. 2 Comparison of an ideal state, a prepared state (with a fidelity of 0.9422 ± 0.0041) and pure states that are constructed from two-photon (MPS2) and three-photon (MPS3) reduced-density matrices; the pure states here and in Fig. 3 (Fig. 4) are obtained using 10,000 MPS-SVT iterations for the two-photon reduced matrices and 2000 MPS-SVT iterations for the three-photon reduced matrices. The real part of these states shows that the ideal state, MPS2 and MPS3 are almost the same; a fidelity of up to 0.9862 ± 0.0012 is found between MPS2 and the ideal state as well as MPS3 and the ideal state (see also Fig. 4)

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To evaluate the effect of the noise during the reconstruction process, we prepare a set of four-photon quantum states ρ₁, ρ₂, ⋯, ρ_m with fidelities ranging from 0.9422 ± 0.0041 to 0.8421 ± 0.0046 by inserting quartz plates of appropriate lengths (30λ ~ 100λ) to act as a dephasing channel. For each state, we repeat the previous measurements and reconstruction processes. The results are shown in Figs. 3 and 4. We evaluate the effect of the noise by examining the fidelity between the reconstructed pure state and the prepared mixed state obtained by the standard quantum state tomography, which is named as the Rfidelity (see Fig. 3), and the fidelity between the reconstructed pure state and the ideal state |Ψ_MPS〉 (see Fig. 4). Fig. 3 clearly shows that Rfidelity tends to 1 along the solid (the dashed) line when the prepared mixed state approaches the ideal pure state, regardless of whether the state is reconstructed from two-photon (blue points) or three–photon (red points) reduced-density matrices. The experimental results agree with the theoretical results (shown as solid and dashed lines in Fig. 3, respectively). In fact, the reconstructed pure state from the noisy reduced-density matrices is very close to the ideal pure state |Ψ_MPS〉 (Fig. 4) (with fidelity above 0.9674 ± 0.0022), indicating that the reconstructed process is robust to the noise [19]. In addition, the saturation of the size of the reduced-density matrices is clearly shown in our experiment. From Figs. 2, 3 and 4, we can see that the reduced density matrices of a larger size cannot help to reconstruct a more precise global state. The size of the reduced matrices is saturated at approximately ⌊log D⌋ + 1, where D = 3 for the ideal state in our experiment, |Ψ_MPS〉. Thus, the two-photon reduced-density matrices are sufficient to reconstruct the global pure state. Our experimental results verify the saturation effect.

Fig. 3 Results from experimental reconstruction: the Y axis corresponds to Rfidelity, which is the fidelity between the reconstructed pure states and the four-photon prepared states obtained using standard quantum tomography; the X axis corresponds to the fidelity of the prepared states; the data points shown in red and blue correspond to the pure states reconstructed from the measured three-photon and two-photon reduced-density matrices, respectively; the solid and dashed lines indicate the pure states reconstructed from the theoretical two-photon and three-photon reduced matrices from dephasing noise on the ideal state, respectively; our experimental results agree with the theoretical results; it shows that the Rfidelity will asymptotically achieve 1 when the prepared state is approaching the ideal state; the error bars in this figure and Fig. 4 are obtained by performing 200 Monte Carlo simulation runs.

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Fig. 4 Fidelity between the reconstructed pure states and the ideal state: the results show that the efficient quantum state tomography is robust to noise; the pure states reconstructed from the reduced matrices of different prepared states are almost the same as the ideal state, and the lowest fidelity of the reconstructed state is 0.9674 ± 0.0022; the saturation of the size of the reduced matrices is also clearly shown.

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5. Conclusion

Our experiment demonstrates that the efficient quantum state tomography of an MPS is valid and robust to the noise. Due to the limitation of the experimental abilities, we just demonstrate the validity and the robustness of the reconstruction method for a very special MPS, which is a permutationally invariant 4-qubit state. However, according to the existing theories, our method can be extended to a large set of states except this special state. More experimental researches should be implemented on the general MPS of the few-body system to test the robustness of the method.

In our current case, the fidelity between the ideal pure state and the prepared state is calculated by the standard quantum tomography. When it comes to a general many-body MPS state, there exists some efficient method [32] to determine the fidelity between the desired pure state and the actual prepared state. In this method [32], only a constant number of the Pauli observables, which are randomly selected through an important-weighting rule, need to be measured repeatedly. The fidelity between the desired state and the actual state can be estimated, up to a constant additive error, by the expectation value of the chosen Pauli observables. This kind of direct fidelity estimation paves a way to experimentally investigate the validity of [19] for multi-qubit state in future. There is still a lot of work to show the dramatic power of the efficient quantum state tomography against the traditional quantum state tomography.

Since the method is efficient both for experimental measurements and for post-processing, it is a very promising technique for experimentally verifying the form of many-body pure states [6] and offers a way of verifying the special character of some exotic many-body states, such as Majorana zero modes [31] using quantum state tomography in a 1D condensed-matter system. It is promising to using this kind of quantum state tomography to the real quantum systems (such as a cold atom system) in the future.

Funding

National Natural Science Foundation of China (grant nos. 11574291, 11274289, 11325419, 11105135, 11474267, 61327901, 61108009 and 61222504); the Fundamental Research Funds for the Central Universities (grant nos. WK2030020019); the Strategic Priority Research Program (B) of the Chinese Academy of Sciences (Grant No. XDB01030300).

Acknowledgments

The authors thank F. L. Xiong for useful discussions.

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Experimental demonstration of efficient quantum state tomography of matrix product states

Abstract

1. Introduction

2. Matrix product states

3. Algorithm for reconstructing an MPS from its reduced density matrices

4. Experimental results

5. Conclusion

Funding

Acknowledgments

References and links

Cited By

Figures (4)

Equations (4)

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