A tailor-made quantum state tomography approach

In the past few years, the race for quantum technologies has favored the realization of large and complex quantum states in various platforms, including superconducting,¹ atomic,^2,3 and photonic systems.^4–9 Such advances are an important resource for the implementation of useful quantum computers and an extraordinary opportunity for fundamental studies in quantum mechanics.

Ideally, one would like to know the complete state of a quantum system, for this is sufficient to compute the value of any other observable. However, as the dimension of the quantum system increases, this task is not trivial at all. For example, consider quantum state tomography (QST),¹⁰ which is arguably the most famous approach for quantum state reconstruction. The idea is to obtain the density matrix of the system from the measurements of a set of observables on a large enough ensemble of identical copies of the state. This method is general and does not make any a priori assumption on the state. However, a system composed of n qudits requires determining the expectation value of at least d²ⁿ observables, that is, the number of independent entries of the density matrix. Such an exponential growth of the number of measurements with the number of qudits can make the experimental implementation of QST unfeasible even for just three or four qudits, depending on the quantum system under investigation.

Over the years, strategies have been developed to decrease the number of required measurements, usually by leveraging certain assumptions about the state or the anticipated outcome. For example, the efficiency of compressed sensing QST depends on the rank of the density matrix, with the state characterization done via a certain number of random Pauli measurements.¹¹ Further improvements in this approach have reduced the number of necessary measurement sets significantly.^12,13 In QST via reduced density matrices, one necessitates the state to be uniquely determined by local reduced density matrices or the assumption that the global state is pure.¹⁴ In Bayesian QST, one defines conditional probabilities starting from measurement results and a suitable a priori distribution over the space of possible states.^15,16 In self-guided QST, one addresses tomography through optimization rather than estimation, iteratively learning the quantum state by treating tomography as a projection measurement optimization problem.^17–19 

Other strategies avoid the reconstruction of the density matrix by focusing only on some relevant information about the state. For instance, in shadow tomography, one can estimate the value of a large number of observables with a few copies of the unknown state.^20–22 Finally, quantum witnesses are useful to verify some crucial properties of a quantum state, such as its degree of entanglement and the kind of quantum correlation.²³ While these last two approaches are extremely useful and applicable to states of large dimensions, it is somewhat disappointing that, after all the efforts to implement a state, one cannot look at it as a whole.

In this work, we describe and analyze threshold Quantum State Tomography (tQST),²⁴ a protocol that allows one to reconstruct the density matrix of a quantum state, even of large dimension, by allowing an efficient trade-off between the number of measurements and the accuracy of the state reconstruction through the presence of a threshold parameter. We show that tQST can drastically reduce the resources required for state reconstruction, but it can also be used to obtain an approximated density matrix by further reducing the number of measurements and the experimental efforts. The protocol can be applied to any quantum system and does not make any assumption about the state to be characterized. First, we outline the tQST procedure and discuss its implementations in the case of n-qubits. Second, we experimentally verify the protocol by performing tQST on systems up to 7 qubits with the IBMQ platform and compare the results with conventional QST. Finally, we draw our conclusions.

In conventional QST, the number of measurements required to reconstruct the density matrix is uniquely determined by its dimension.¹⁰ Specifically, a system of n qubits requires at least 4ⁿ measurements, that is, the number of independent parameters of the state density matrix. These measurements can be performed in an arbitrary sequence, and the results are finally combined to obtain the state density matrix through maximum likelihood estimation or other approaches.^25–27 

Here, we begin by noting that a density matrix, denoted as ρ, is required to have a unit trace, be Hermitian, and positive semi-definite. Specifically, the latter property entails that any element ρ_ij of a physical density matrix ρ must fulfill the condition $|ρij|≤ρiiρjj$⁠. Thus, measuring the diagonal elements of the density matrix (i.e., projecting on the states of the computational basis) provides some information on the off-diagonal terms. Indeed, if ρ_ii is found to be zero, then one can immediately set to zero all the elements of the ith row and column of ρ. Similarly, if ρ_ii and ρ_jj are different from zero but small compared to the other diagonal elements, one knows that the modulus of ρ_ij will be small too.

These considerations are at the basis of tQST, whose protocol is illustrated in Fig. 1. First, one measures the diagonal elements {ρ_ii} of the density matrix, which are directly accessible by projecting on the chosen computational basis. Second, one chooses a threshold t and, by exploiting the information on {ρ_ii}, identifies those ρ_ij for which $ρiiρjj≥t$⁠. Third, one constructs a proper set of observables associated with these ρ_ij (see subsection n-qubit projectors) and performs only those measurements. Finally, one uses these results to reconstruct the density matrix, for example, using maximum-likelihood estimation. We stress that maximum likelihood estimation is not the only possible choice, and one can consider other approaches, such as linear inversion, Bayesian mean estimation,²⁸ or linear regression estimation.²⁹ 

In tQST, the information achieved by measuring the diagonal terms of ρ is immediately used to decide the subsequent measurements to be performed, by choosing to neglect the terms of ρ that in modulus are smaller than a certain threshold. Unlike conventional QST, the resources necessary to reconstruct the state are not uniquely determined by the dimension of the quantum system but can be controlled with the threshold t. For example, if one sets t = 0, all the elements of ρ are considered, and the protocol reduces to conventional QST. On the contrary, for t > 0, the protocol may require fewer measurements than those needed with conventional QST and, in any case, no more than them. Importantly, one does not make any a priori assumption on the state or the result of the characterization. It should be noted that unlike adaptive approaches,^12,13 in which the necessary measurement sets cannot be known a priori, for each one is chosen based on the outcomes of the previous measurement set, in tQST, the projectors can all be determined once the system is measured in the chosen computational basis. In this respect, we hasten to emphasize that, once a threshold value is chosen, the protocol does not simply reduce to measuring a subspace of the Hilbert space of the whole system.

The threshold t determines the amount of information one is willing to trade in exchange for fewer measurements. This has several consequences. First, one may be able to reconstruct the entire density matrix of a large state by reducing the amount of measurements to a level compatible with the available experimental resources. Second, fewer measurements can give a significant advantage in terms of storing and handling experimental data. Third, one may be able to avoid useless measurements and increase the integration time for the remaining measurements, leading to an improvement in the signal-to-noise ratio. The amount of information obtainable in the characterization of a quantum state is always limited, for example by noise or the finite precision of the experimental setup. Such experimental constraints de facto bound the accuracy with which ρ can be determined, even for traditional QST. Thus, although one may naively expect that reducing the number of measurements will decrease the quality of the results, as we shall see in the following, a wise choice of t can still guarantee practically reaching the best achievable result while requiring fewer resources.

In Fig. 2, we show the simulated reconstruction of the density matrix via tQST with maximum likelihood estimation of several W states for different numbers of qubits ranging from 4 to 7.³⁰ Even for the case of 7 qubits, in which traditional QST would have required 16 384 measurements, one can reconstruct ρ with a fidelity of about 99% with the target state by implementing only 170 measurements. We stress here that the dramatic reduction in the number of measurements is not simply given by the choice of a particular state, but rather by the employed computational basis, which affects the final state representation. As in compressed sensing tomography or similar approaches, sparse matrices are usually easier to reconstruct, because most of the information is contained in fewer elements of the density matrix. Yet, for sparse matrices, the advantage of tQST can be significantly larger than with other approaches. Take, for example, the 7-qubit state reconstructed in Ref. 31 via compressed sensing QST, in which the characterization required 16 256 projective measurements. While this number is significantly smaller than 6ⁿ, which is the number of observables of the typical tomographically overcomplete set, this is still 99% of the 4ⁿ observables that are strictly necessary.¹⁰ On the contrary, the very same state could be reconstructed via tQST by performing only 184 measurements, which is about 1% of those required by compressed sensing (see the supplementary material). We stress that tQST can also be applied to matrices that are not sparse, where the threshold t will determine both the number of measurements and the reliability of the reconstruction.

Our approach can be extended to systems composed of n qudits (see the supplementary material). In the case of qubits, we developed a Python package, which implements tQST for an arbitrary number of qubits that are freely available to users on GitHub.³² 

We experimentally demonstrate tQST by using the IBMQ processor lagos,^33,34 which allows one to prepare states with up to 7 superconducting qubits. In our implementation, each time we program the system to generate a target quantum state by constructing the corresponding quantum circuit. We first reconstruct the state density matrix by performing conventional QST as implemented by IBMQ, which uses linear inversion on the outcomes of an overcomplete set of 6ⁿ observables. This process generally yields a non-positive reconstructed state, which is subsequently rescaled to be positive semi-definite.³⁵ We then reconstruct the same state by using tQST, where the threshold t is chosen by considering the typical signal-to-noise ratio (SNR) of the system to avoid unnecessary measurements (Sec. IV B). In this case, we use a maximum likelihood estimation to obtain the (positive semi-definite) density matrix from the measured expectation values (Sec. IV D). To compare the results, we compute the fidelities between the target state and the reconstructed ones and also that between the two reconstructed states. This choice is motivated by the fact that while not exhaustive, fidelity remains a widely recognized metric.

In our analysis, we generated 300 random states for a number of qubits from 4 to 7 and sorted them according to the diagonal filling, i.e., the percentage of expected non-vanishing diagonal elements. In Fig. 3, we show representative results for filling percentages of 25% (a) and (d), 50% (b) and (e), and 75% (c) and (f) for the case of 4 and 7 qubits (other representative results for 5 and 6 qubits are shown in Fig. 1 of the supplementary material). Each panel of the figure shows the circuit to generate the state, the density matrix reconstructed with conventional QST, and the one obtained with tQST. In this last case, we also report the value of the threshold t and the corresponding number of measurements used in the reconstruction. Since we choose t according to the typical SNR of the system, such a number is also related to the sparsity of the state representation. For the cases of 4, 5, and 6 qubits, we observe up to a 100-fold reduction in the number of required measurements compared to IBMQ-QST. For the case of 7 qubits, such a reduction is $∼300$ for the case of Fig. 3(d).

As evident from the state representation, despite the significant difference in the number of measurements, the states reconstructed with IBMQ-QST and tQST are very similar. A more quantitative analysis is obtained by calculating the fidelity between the two reconstructed states, which in all cases is about 90% or above, as shown in Fig. 4. The same figure also reports the fidelity of the tQST-reconstructed state with the target one. Remarkably, this is always comparable (within errors) to the fidelity between the IBMQ-QST reconstructed state and the target one (see Table S1 of the supplementary material). Thus, no advantage is obtained with IBMQ-QST by performing more measurements than those set by the threshold in tQST. This suggests that when the threshold is determined by the SNR of the system, the tQST protocol can extract all the amount of information accessible with conventional QST, yet with a smaller number of measurements.

The IBMQ processor lagos limits our analysis to 7 qubits. Yet, it is interesting to further investigate the performance of tQST by extrapolating the analysis to a larger number of qubits. In this analysis, we considered simulated data with a SNR analogous to that of the IBMQ system and followed the same strategy for the threshold choice. We considered the case of W states and increased the number of qubits up to 14, limited now only by our hardware. In Table I, we report the number of qubits, the threshold t, the number of measurements, and the fidelity with respect to the target state. In all the cases, we obtained fidelities exceeding 90%. We stress that in the cases of 14 qubits, the 16 556 measurements required by tQST make the reconstruction in principle experimentally accessible today. On the contrary, conventional QST would necessitate at least 4¹⁴ = 268, 435, 456 measurements that, at the moment, appear as a prohibitive number.

We have proposed and implemented a protocol for QST that allows one to reconstruct the density matrix of any quantum state with a number of measurements that can be considerably smaller than that required by conventional QST. Such an approach does not make any a priori assumptions on the state and employs a threshold t to control the amount of information used in the state reconstruction. The presence of such a threshold allows one to trade the minimum fidelity with which the state can be reconstructed with the amount of resources to perform the characterization, thus reducing the number of measurements significantly. In addition, the threshold can be set to take into account the experimental limitations that may lead to unnecessary measurements. Our protocol was implemented on the IBMQ system lagos to characterize random states up to 7 qubits. In all the considered cases, the fidelity achieved with tQST was compatible within the experimental uncertainty with that obtained by using conventional QST but with a smaller amount of measurements (in some cases $∼300$ times smaller). This suggests that our protocol is able to efficiently access all the information that can be extracted from the system. Finally, by using synthetic data, we pushed the approach to our computational limit and performed the characterization of W states up to 14 qubits, reaching a fidelity larger than 90% with only $∼16 000$ expectation values, four orders of magnitudes less than what would be required by conventional QST.

Our protocol is a flexible and practical approach for the full characterization of large quantum systems, including those based on atoms or photons. For example, the reconstruction of the 8-qubit state that was done in Ref. 2 took hours and required days to handle the outcomes of some 656 000 measurements, but our approach would require only 312 measurements, thus reducing the experimental efforts significantly. Similarly, in the case of photon-based systems⁴ for which the detection rate is often limited by loss, our tomographic approach is expected to allow for the full reconstruction of states exceeding 10 photons. For these reasons, we believe that tQST will be particularly useful to the whole quantum community.

The recursive structure outlined in Eq. (8) can be leveraged to reduce the necessity of generating the entire set of projectors upfront. Instead, we can generate projectors on-demand, specifically for elements of the density matrix that need to be measured to achieve faithful reconstruction given a certain threshold.

To this end, we divide π_n into four quadrants, with “1” referring to the upper-left quadrant and “4” indicating the lower-right quadrant. Each of the quadrants 2 and 3 is further divided into an upper (“u”) and a lower (“l”) triangular part. The real part of the elements along the diagonal is assigned to the u portion, while the imaginary part is assigned to the l portion. This subdivision is pictorially represented in Fig. 5.

To determine the projector corresponding to the density-matrix element ρ_i,j, we initially locate it within π_n, which immediately determines the projector associated with the first qubit, that is, |H⟩, |D⟩, |R⟩, or |V⟩ according to its position: 1, 2u, 2l, or 4, respectively. We then continue splitting the quadrant where the element is found until we reach a resulting quadrant size of 2 × 2. At each splitting step, if the element falls into quadrant 1 or 4, the projector associated with the next qubit is |H⟩ (for 1) or |V⟩ (for 4). Conversely, if it falls into quadrant 2 or 3, the projector choice depends on its position in the previous splitting step. For an element in an upper quadrant, we select |D⟩; and, for an element in a lower quadrant, we select |R⟩ unless the previous quadrant was either 2l or 3l, in which case the choice is reversed. Table II summarizes these steps.

As an illustration, Fig. 5 shows the determination of two projectors for a 3- and a 4-qubit system. More specifically, in the 3-qubit case, we consider the matrix element $ρ3,5im$⁠; its successive locations in π₃ are described by the sequence “2l3l4,” which, according to Table II, corresponds to the local projector |RDV⟩. The projector associated with $ρ4,9re$ in a 4-qubit system is instead determined by the locations in π₄ described by the sequence “2l3u12u”; using Table II, one then finds |RRHD⟩.

The selection of an appropriate threshold value is dependent on the specific physical system used to implement the qubits (e.g., noise level), the amount of available resources (e.g., time requirements), and the desired quantum state to be generated.

The supplementary material contains a more detailed analysis of the states reconstructed on the IBM quantum processor, a comparison between tQST and the compressed sensing approach, and an outline on the extension of tQST to qudit systems.

We thank Professor Daniel F. V. James of the University of Toronto for useful discussions.

This work was supported by PNRR MUR, Project No. PE0000023-NQSTI.

The authors have no conflicts to disclose.

D. Binosi: Data curation (equal); Formal analysis (equal); Investigation (equal); Methodology (equal); Software (equal); Writing – original draft (equal); Writing – review & editing (equal). G. Garberoglio: Data curation (equal); Formal analysis (equal); Investigation (equal); Methodology (equal); Software (equal); Writing – original draft (equal); Writing – review & editing (equal). D. Maragnano: Conceptualization (equal); Data curation (equal); Formal analysis (equal); Investigation (equal); Methodology (equal); Software (supporting); Writing – original draft (equal); Writing – review & editing (equal). M. Dapor: Formal analysis (equal); Funding acquisition (equal); Investigation (equal); Methodology (equal); Project administration (equal); Writing – original draft (equal); Writing – review & editing (equal). M. Liscidini: Conceptualization (equal); Formal analysis (equal); Funding acquisition (equal); Investigation (equal); Methodology (equal); Project administration (equal); Writing – original draft (equal); Writing – review & editing (equal).

The data that support the findings of this study are available from the corresponding author upon reasonable request.