1 Introduction

Quantum state tomography is a fundamental important step in quantum information processing (Nielsen and Chuang 2000; Paris and Řeháček 2004). In general, it aims at finding the underlying density matrix which describes the given state of a physical quantum system. This task is done by utilizing the results of measurements performed on repeated state preparations (Nielsen and Chuang 2000).

Bayesian methods have been recognized as a powerful paradigm for quantum state tomography (Blume-Kohout 2010), that deal with uncertainty in meaningful and informative ways and are the most accurate approach with respect to the expected error (operational divergence) even with finite samples. Several studies have been conducted: for example, the papers Bužek et al. (1998) and Baier et al. (2007) performed numerical comparisons between Bayesian estimations with other methods on simulated data; algorithms for computing Bayesian estimators have been discussed in Kravtsov et al. (2013), Ferrie (2014), Kueng and Ferrie (2015), Schmied (2016) and Lukens et al. (2020).

Pseudo-Bayesian method for quantum tomography, introduced in Mai and Alquier (2017), proposes novel approaches for this problem with several attractive features. Importantly, a novel prior distribution for quantum density matrix is introduced based on spectral decomposition parameterization (inspired by the priors used for low-rank matrix estimation, e.g., Mai and Alquier (2015) and Cottet and Alquier (2018)). This prior can be easily used in any dimension and is found to be significantly more efficient to sample from and evaluate than the Cholesky approach in Struchalin et al. (2016), Zyczkowski et al. (2011) and Seah et al. (2015), see Lukens et al. (2020) for more details. By replacing the likelihood with a loss function between a proposed density matrix and experimental data, the paper Mai and Alquier (2017) presents two different estimators: the prob-estimator and the dens-estimator.

However, the reference Mai and Alquier (2017) proposed simply to compute approximately these two Pseudo-Bayesian estimators by naive Metropolis-Hastings (MH) algorithms which are computationally very slow for high-dimensional systems. Recently, a faster and more efficient sampling method has been proposed for the dens-estimator, see Lukens et al. (2020). However, we would like to note that the prob-estimator is shown in Mai and Alquier (2017) to reach the best known up-to-date rate of convergence (Butucea et al. 2015) while the theoretical guarantee for the dens-estimator is far less satisfactory. Moreover, it is also shown in their simulations that the prob-estimator yields better results compared to the dens-estimator.

In this paper, we present a novel efficient adaptive Metropolis-Hastings implementation for the prob-estimator. This adaptive implementation is based on considering the whole density matrix as a parameter that needs to be sampled at a time. Moreover, an adaptive proposal is explored based on the “preconditioned Crank-Nicolson” (Cotter et al. 2013) sampling procedure that can eliminate the “curse of dimensionality”, which is the case for quantum state tomography where the dimension increases exponentially. Further speeding up by using subsampling MCMC approach is also explored in our work.

Through simulations, it is shown that our implementation is computationally significantly faster than the naive MH algorithm in Mai and Alquier (2017). More specifically, for example as in a system of 6 qubits, our algorithm is around 115 times faster than the naive MH algorithm in Mai and Alquier (2017). While in term of accuracy, we show that our algorithms returns similar results with less variation.

The rest of the paper is organized as follow. In Sect. 2, we provide the necessary background and the statistical model for the problem of quantum state tomography. In Sect. 3, we recall the Pseudo-Bayesian approach and the prior distribution. Section 4 presents our novel adaptive MCMC implementation for the Pseudo-Bayesian estimator. Simulation studies are presented in Sect. 5. Conclusions are given in Sect. 6.

2 Background

2.1 The quantum state tomography problem

Hereafter, we only provide the necessary background on quantum state tomography (QST) required for the paper. We would like to remind that a very nice introduction to this problem, from a statistical perspective, can be found in Artiles et al. (2005). Here, we have opted for the notations used in reference Mai and Alquier (2017).

Mathematically speaking, a two-level quantum system of n-qubits is characterized by a \( 2^{n}\times 2^{n} \) density matrix \(\rho \) whose entries are complex, i.e. \( \rho \in {\mathbb {C}}^{2^{n}\times 2^{n} } \). For the sake of simplicity, put \(d=2^n\), so \(\rho \) is a \(d\times d\) matrix. This density matrix must be

  • Hermitian: \(\rho ^\dagger =\rho \) (i.e. self-adjoint),

  • positive semi-definite: \(\rho \succcurlyeq 0\),

  • normalized: \(\mathrm{Trace}(\rho )=1\).

In addition, physicists are especially interested in pure states and that a pure state \( \rho \) can be defined in addition by \(\mathrm{rank}(\rho )=1\). In practice, it often makes sense to assume that the rank of \(\rho \) is small (Gross et al. 2010; Gross 2011; Butucea et al. 2015).

The goal of quantum tomography is to estimate the underlying density matrix \( \rho \) using measurement outcomes of many independent and identically systems prepared in the state \( \rho \) by the same experimental devices.

For a qubit, it is a standard procedure to measure one of the three Pauli observables \(\sigma _x, \, \sigma _y, \, \sigma _z\). The outcome for each will be 1 or \( -1 \), randomly (the corresponding probability is given in (1) below). As a consequence, with a n-qubits system, there are \(3^n\) possible experimental observables. The set of all possible performed observables is

$$\begin{aligned} \{\sigma _{{\mathbf {a}}} = \sigma _{{a}_1} \otimes \cdots \otimes \sigma _{{a}_n}; \, {\mathbf {a}} = (a_1,\ldots ,a_n) \in {\mathcal {E}}^n := \{x,y,z\}^{n}\}, \end{aligned}$$

where vector \({\mathbf {a}} \) identifies the experiment. The outcome for each fixed observable setting will be a random vector \( {\mathbf {s}} = (s_1, \ldots , s_n) \in \{-1,1\}^{n} \), thus there are \( 2^n \) outcomes in total.

Denote \(R^{{\mathbf {a}}}\) a random vector that is the outcome of an experiment indexed by \({\mathbf {a}}\). From the Born’s rule (Nielsen and Chuang 2000), its probability distribution is given by

$$\begin{aligned} p_{{\mathbf {a}},{\mathbf {s}}} := {\mathbb {P}} (R^{\mathbf {a}}= {\mathbf {s}}) = \mathrm{Trace} \left( \rho \cdot P_{{\mathbf {s}}}^{{\mathbf {a}}} \right) , \forall {\mathbf {s}} \in \{-1,1\}^{n}, \end{aligned}$$
(1)

where \( P_{{\mathbf {s}}}^{{\mathbf {a}}} := P_{s_1}^{a_{1}}\otimes \dots \otimes P_{s_n}^{a_n}\) and \(P_{s_i}^{a_i}\) is the orthogonal projection associated to the eigenvalues \( s_i\in \{ \pm 1 \} \) in the diagonalization of \( \sigma _{a_i ; , a_i\in \{x,y,z\} } \) – that is \( \sigma _{a_i} = P^{a_i}_{+1} -P^{a_i}_{-1} \).

Statistically, for each experiment \( {\mathbf {a}}\in {\mathcal {E}}^n\), the experimenter repeats m times the experiment corresponding to \({\mathbf {a}}\) and thus collects m independent random copies of \(R^{\mathbf {a}}\), say \(R^{\mathbf {a}}_1,\dots ,R^{\mathbf {a}}_m\). As there are \(3^n\) possible experimental settings \( {\mathbf {a}}\), we define the quantum sample size as \( N:=m\cdot 3^n \). We will refer to \((R^{\mathbf {a}}_i)_{i\in \{1,\dots ,m\},{\mathbf {a}}\in {\mathcal {E}}^n}\) as \({\mathcal {D}}\) (for data). Therefore, quantum state tomography is aiming at estimating the density matrix \( \rho \) based on the data \({\mathcal {D}}\).

2.2 Popular estimation methods

Here, we briefly recall three classical major approaches that have been adopted to estimate \( \rho \), which are: linear inversion, maximum likelihood and Bayesian inference.

2.2.1 Linear inversion

The first and simplest method considered in quantum information processing is the ’tomographic’ method, also known as linear/direct inversion (Vogel and Risken 1989; Řeháček et al. 2010). It is actually the analogue of the least-square estimator in the quantum setting. This method relies on the fact that measurement outcome probabilities are linear functions of the density matrix.

More specifically, let us consider the empirical frequencies

$$\begin{aligned} {\hat{p}}_{{\mathbf {a}},{\mathbf {s}}} = \frac{1}{m}\sum _{i=1}^m {\mathbf {1}}_{\{R_i^{\mathbf {a}}={\mathbf {s}}\}}. \end{aligned}$$
(2)

It is noted that \( {\hat{p}}_{{\mathbf {a}},{\mathbf {s}}}\) is an unbiased estimator of the underlying probability \( p_{{\mathbf {a}},{\mathbf {s}}} \) in (1). Therefore, the inversion method is based on solving the linear system of equations

$$\begin{aligned} \left\{ \begin{array}{l} {{\hat{p}}}_{{\mathbf {a}},{\mathbf {s}}} = \mathrm{Trace} \left( {\hat{\rho }} \cdot P_{{\mathbf {s}}}^{{\mathbf {a}}} \right) , \\ {\mathbf {a}}\in {\mathcal {E}}^n,\quad {\mathbf {s}} \in \{-1,1\}^{n}. \end{array} \right. \end{aligned}$$
(3)

As mentioned above, the computation of \({\hat{\rho }}\) is quite clear and explicit formulas are classical that can be found for example in Alquier et al. (2013). While straightforward and providing unbiased estimate (Schwemmer et al. 2015), it tends to generate a non-physical density matrix as an output (Shang et al. 2014): positive semi-definiteness cannot easily be satisfied and enforced.

2.2.2 Maximum likelihood

A popular approach in QST in recent years is the maximum likelihood estimation (MLE). MLE aims at finding the density matrix which is most likely to have produced the observed data \({\mathcal {D}}\):

$$\begin{aligned} \rho _{MLE} = \arg \max L(\rho ;\;{\mathcal {D}}) \end{aligned}$$

where \( L(\rho ;{\mathcal {D}}) \) is likelihood, the probability of observing the outcomes given state \( \rho \), as defined by some models (Hradil et al. 2004; James et al. 2001; Gonçalves et al. 2018). However, it has some critical problems, detailed in Blume-Kohout (2010), including a huge computational cost. Moreover, it is a point estimate which does not account the level of uncertainty in the result.

Furthermore, these two methods (Linear inversion and MLE) can not take advantage of a prior knowledge where a system is in a state \(\rho \) for which some additional information is available. More particularly, it is noted that physicists usually focus on so-called pure states, for which \(\mathrm{rank}(\rho )=1\).

2.2.3 Bayesian inference

Starting receiving attention in recent years, Bayesian QST had been shown as a promising method in this problem (Blume-Kohout 2010; Bužek et al. 1998; Baier et al. 2007; Lukens et al. 2020). Through Bayes’ theorem, experimental uncertainty is explicitly accounted in Bayesian estimation. More specifically, suppose a density matrix \( \rho \) is parameterized by \( \rho (x) \) for some x, Bayesian inference is carried out via the posterior distribution

$$\begin{aligned} \pi ( \rho (x) | {\mathcal {D}} ) \propto L(\rho (x);{\mathcal {D}}) \pi (x), \end{aligned}$$

where \( L(\rho (x);{\mathcal {D}}) \) is the likelihood (as in MLE) and \( \pi (x) \) is the prior distribution. Using the posterior distribution \( \pi (\rho ( x) | {\mathcal {D}} ) \), the expectation value of any function of \( \rho \) can be inferred, e.g. the Bayesian mean estimator as \( \int \rho (x) \pi (\rho ( x) | {\mathcal {D}}) dx \).

Although recognized as a powerful approach, the numerical challenge of sampling from a high-dimensional probability distribution prevents the widespread use of Bayesian methods in the physical problem.

2.2.4 Other approaches

Several other methods have also recently introduced and studied. The reference Cai et al. (2016) proposed a method based on the expansion of the density matrix \( \rho \) in the Pauli basic. Some rank-penalized approaches were studied in Guţă et al. (2012) and Alquier et al. (2013). A thresholding method is introduced in Butucea et al. (2015).

3 Pseudo-Bayesian quantum state tomography

3.1 Pseudo-Bayesian estimation

Let us consider the pseudo-posterior, studied in Mai and Alquier (2017), defined by

$$\begin{aligned} {\tilde{\pi }}_{\lambda }(\mathrm{d}\nu ) \propto \exp \left[ -\lambda \ell (\nu ,{\mathcal {D}}) \right] \pi (\mathrm{d}\nu ), \end{aligned}$$

where \(\exp \left[ -\lambda \ell (\nu ,{\mathcal {D}}) \right] \) is the pseudo-likelihood that plays the role of the empirical evidence to give more weight to the density \(\nu \) when it fits the data well; \(\pi (\mathrm{d}\nu )\) is the prior given in Sect. 3.2; and \(\lambda >0\) is a tuning parameter that balances between evidence from the data and prior information.

Taking, with \( {\hat{p}}_{{\mathbf {a}},{\mathbf {s}}} \) given in (2),

$$\begin{aligned} \ell (\nu ,{\mathcal {D}}) := \ell ^{prob}(\nu ,{\mathcal {D}}) = \sum _{{\mathbf {a}}\in {\mathcal {E}}^n} \sum _{{\mathbf {s}}\in \{-1,1\}^n} \left[ \mathrm{Tr}(\nu P_{\mathbf {s}}^{\mathbf {a}}) - {\hat{p}}_{{\mathbf {a}},{\mathbf {s}}} \right] ^2, \end{aligned}$$

the “prob-estimator” in Mai and Alquier (2017) is defined as the mean estimator of the pseudo-posterior:

$$\begin{aligned} {\tilde{\rho }}^{prob}_{\lambda } = \int \nu \exp \left[ -\lambda \ell ^{prob}(\nu ,{\mathcal {D}}) \right] \pi (\mathrm{d}\nu ). \end{aligned}$$
(4)

This estimator also referred to, in statistical machine learning, as Gibbs estimator, PAC-Bayesian estimator or EWA (exponentially weighted aggregate) (Catoni 2007; Dalalyan and Tsybakov 2008).

For the sake of simplicity, we use the shortened notation \( p_\nu := [\mathrm{Tr}(\nu P_{\mathbf {s}}^ {\mathbf {a}})]_{{\mathbf {a}},{\mathbf {s}}} \) and \( {\hat{p}} := [{\hat{p}}_{{\mathbf {a}}, {\mathbf {s}}}]_{{\mathbf {a}},{\mathbf {s}}} \) then

$$\begin{aligned} \ell ^{prob}(\nu ,{\mathcal {D}}) = \Vert p_\nu - {\hat{p}} \Vert ^2_F \end{aligned}$$

(\( \Vert \cdot \Vert _F \) is the Frobenius norm). Clearly, we can see that this distance measures the difference between the probabilities given by a density \( \nu \) and the empirical frequencies in the sample. Here, the readers could see that the tuning parameter \( \lambda \) is used to control the difference between the empirical frequencies and the hypothetical one from the prior distribution. We remind the reader that the matrix \( [{\hat{p}}_{{\mathbf {a}}, {\mathbf {s}}}]_{{\mathbf {a}},{\mathbf {s}}} \) is of dimension \( 3^n \times 2^n \).

Remark 1

This kind of pseudo-posterior is an increasingly popular approach in Bayesian statistics and machine learning, see for example Bissiri et al. (2016), Mai (2021b), Grünwald and Van Ommen (2017), Catoni (2007), Mai (2022), Alquier et al. (2016b), Mai (2021a) and Bégin et al. (2016), for models with intractable likelihood or for misspecification models.

3.2 Prior distribution for quantum density matrix

The prior distribution employed in Mai and Alquier (2017) can be expressed as follow: the \( d\times d \) density matrix \( \rho \) can be parameterized by d non-negative real numbers \(y_i \) and d complex column vectors of length d, \(z_i \). Put \(x = \left\{ y_1, \ldots , y_d, z_1 , \ldots , z_d \right\} \), then the density matrix is

$$\begin{aligned} \rho (x) = \sum _{i=1}^d \dfrac{y_i}{ \sum _\ell y_\ell } \dfrac{z_i z_i^\dagger }{\Vert z_i\Vert ^2} , \end{aligned}$$
(5)

with the prior distribution for x as

$$\begin{aligned} \pi (x) \propto \prod _{i =1}^d y_i^{\alpha - 1} e^{-y_i} e^{-\frac{1}{2}z_i^\dagger z_i} \end{aligned}$$
(6)

where the weights are being treated as Gamma-distributed random variables \( Y_i \overset{i.i.d.}{\sim } \Gamma (\alpha ,1) \), and the vectors \( z_i \) are standard-normal complex Gaussian distributed \( Z_i \overset{i.i.d.}{\sim } {{\mathcal {C}}}{{\mathcal {N}}} (0, I_d) \).

The tuning parameter \( \alpha \) in (6) allows the user to favor low-rank or high-rank density matrices which are corresponding to pure or mixed states, respectively. More particularly, the normalized random variables \( Y_i /(\sum Y_j) \) with \( Y_i \overset{i.i.d.}{\sim } \Gamma (\alpha ,1) \) follows a Dirichlet distribution \( \mathrm{Dir} (\alpha ) \) which ensures both normalization and non-negativity. An \( \alpha <1 \) promotes sparse draws and thus purer states, while \( \alpha =1 \) returns a fully uniform prior on all physically realizable states.

Remark 2

It is noted that this parameterization satisfies all physical conditions for the density matrix, details can be found in Mai and Alquier (2017). Moreover, this parameterization have been shown to be significantly more efficient to sample from and to evaluate than the Cholesky approach in references Struchalin et al. (2016), Zyczkowski et al. (2011) and Seah et al. (2015), see Lukens et al. (2020) for details.

Remark 3

The theoretical guarantees for the “prob-estimator" in (4) are validated only for \(0< \alpha \le 1\). More specifically, the prob-estimator satisfies (up to a multiplicative logarithmic factor) that \( \Vert {\tilde{\rho }}^{prob}_{\lambda ^*} - \rho ^0 \Vert _F^2 \le c 3^n \mathrm{rank}(\rho ^0)/N \) which is the best known up-to-date rate in the problem of quantum state estimation (Butucea et al. 2015), where c is a numerical constant and \(\lambda ^* = m/2\).

4 A novel efficient adaptive MCMC implementation

Appropriately, the prob-estimator requires an evaluation of the integral (4) which is numerically challenging due to its sophisticated features and high dimensionality. A first attempt has been done in Mai and Alquier (2017) is to use a naive Metropolis-Hastings (MH) algorithm where the authors iterate between a random walk MH for \( \log (y_i)\) and an independent MH for \(z_i\). Typically, the approach is designed to obtain T samples \( x^{(1)}, \ldots , x^{(T)} \) as a consequence the integral (4) can be approximated as

$$\begin{aligned} {\hat{\rho }}^{\mathrm{MH}} \approx \frac{1}{T}\sum _{t=1}^T\rho (x^{(t)}). \end{aligned}$$

However, as also noted in the reference Mai and Alquier (2017), their proposed algorithm can run into slow convergence and can be arbitrarily slow as the system dimensionality increases. In this paper, we propose a novel efficient MCMC algorithm for the prob-estimator through exploring the adaptive proposal and subsampling scheme.

4.1 A preconditioned Crank-Nicolson adaptive proposal

Borrowing motivation from the recent work in Lukens et al. (2020) that proposes an efficient sampling procedure for Bayesian quantum state estimation (which improve the computation of the “dens-estimator” in Mai and Alquier (2017) only), we introduce an efficient adaptive Metropolis-Hastings implementation for the prob-estimator in Mai and Alquier (2017). We remind that the prob-estimator shows better performance than the dens-estimator both in theory and simulations.

Specifically, we propose to use a modification of random-walk MH by scaling the previous step before adding a random move and generating the proposal \( z' \). Following Cotter et al. (2013) who introduced an efficient MCMC approach eliminating the “curse of dimensionality”, termed as “preconditioned Crank-Nicolson”, we use the proposal for \(z_j\) as

$$\begin{aligned} z_{j}^{\prime } = \sqrt{1-\beta _z^2} z_{j}^{(k)} + \beta _z\varvec{\xi }_j , \quad \varvec{\xi }_j {\mathop {\sim }\limits ^{\text {i.i.d.}}} {{\mathcal {C}}}{{\mathcal {N}}}(0,I_d) \end{aligned}$$

where \( \beta _z \in (0,1) \) is a tuning parameter. This proposal is a scaled, by the factor \( \sqrt{1-\beta _z^2} \), random walk that results in a slightly simpler acceptance probability. Unlike the independent proposal in Mai and Alquier (2017) (with \( \beta _z=1 \)) where the acceptance probability can vary substantially, this kind of adaptive proposal allows one to control the acceptance rate efficiently. For \( \beta _y \in (0,1) \), we slightly modify the proposal for y from Mai and Alquier (2017) (with \( \beta _y =1 \)) as

$$\begin{aligned} y_j^\prime = y_j^{(j)} e^{\beta _y\eta _j} , \quad \eta _j{\mathop {\sim }\limits ^{\text {i.i.d.}}} \text {Uniform}(-0.5,0.5) . \end{aligned}$$

The acceptance ratio \( \min \{1, A(x' | x^{(k)}) \} \) are followed from the standard form for MH (Robert and Casella 2013). Let \( p(x' | x^{(k)} ) \) denote the proposal density, we have

$$\begin{aligned} A(x' | x^{(k)}) = \dfrac{ {\tilde{\pi }}(\rho (x')) }{ {\tilde{\pi }}(\rho (x^{(k)})) } \dfrac{ p(x^{(k)} | x') }{ p(x' | x^{(k)} ) }, \end{aligned}$$

where

$$\begin{aligned} \log A( x^\prime | x^{(k)}) = \log L_D(x^\prime ) - \log L_D(x^{(k)}) + \sum _{j=1}^d \left[ \alpha \log y_j^\prime - y_j^\prime -\alpha \log y_j^{(k)} + y_j^{(k)} \right] . \end{aligned}$$

4.2 Speeding up by subsampling

We remind that the log (pseudo-)likelihood, \( \log L_D(x) = - \lambda \Vert p_\nu - {\hat{p}} \Vert ^2_F \), is the Frobenius norm of a matrix of dimension \( 3^n \times 2^n \) and thus for large n it will be very costly to evaluate at each iteration. For example, with \( n=7 \), this matrix is of dimension \( 2187\times 128 \). Therefore, we propose to evaluate a random subset of this matrix at each iteration. More precisely, at each iteration, we draw uniformly at random a subset \( \Omega \) of indices of the \( 3^n \times 2^n \) matrix. Then, the log pseudo-likelihood, \( \log L_D(x) \), is approximated by

$$\begin{aligned} \log L_{\Omega }(x) := \sum _{ ({\mathbf {a}} , {\mathbf {s}}) \in \Omega \subset {\mathcal {E}}^n\times \{-1,1\}^n} \left[ \mathrm{Tr}(\nu P_{\mathbf {s}}^{\mathbf {a}}) - {\hat{p}}_{{\mathbf {a}},{\mathbf {s}}} \right] ^2, \end{aligned}$$

As a consequence, the acceptance rate corresponding with this subsampling is denoted by \( A_\Omega ( x^\prime | x^{(k)}) \). It is noted that this kind of using subsampling to speeding up MCMC algorithms is becoming popular in the computational statistics community, see for example Quiroz et al. (2018b), Maire et al. (2019), Quiroz et al. (2018a) and Alquier et al. (2016a).

The details of our novel adaptive MH is given in Algorithm 1.

figure a

5 Numerical studies

5.1 Simulations setups and details

To assess the performance of our new proposed algorithm, a series of experiments were conducted with simulated tomographic data. More particularly, we consider the following setting for choosing the true density matrix, with \( n=2,3,4, \) (\(d=4,8,16 \)):

  • Setting 1: we consider the ideal entangled state which is characterized by a rank-2 density matrix that

    $$\begin{aligned} \rho _{rank-2} = \frac{1}{2}\psi _1 \psi _1^{\dagger } + \frac{1}{2}\psi _2 \psi _2^{\dagger } \end{aligned}$$

    with \( \psi _1 = u /\Vert u\Vert \) and \( u = (u_1, \ldots , u_{d/2}, 0,\ldots ,0 ), u_1 = \ldots = u_{d/2} =1 \); \( \psi _2 = v /\Vert v\Vert \) and \( v = (0,\ldots ,0, v_{d/2 +1}, \ldots , v_d ), v_{d/2} = \ldots = v_d = 1 \).

  • Setting 2: a maximal mixed state (rank-d) that is

    $$\begin{aligned} \rho _{mixed} = \sum _{i=1}^d \frac{1}{d}\psi _i \psi _i^{\dagger }, \end{aligned}$$

    with \( \psi _i \) are normalized vectors and independently simulated from \( {{\mathcal {C}}}{{\mathcal {N}}}(0,I_d) \).

The experiments are done following Sect. 2 for \( m=1000 \). The prob-estimator is employed with \( \lambda = m/2 \) and a prior with \( \alpha =1 \) which are theoretically guaranteed from Theorem 1 in reference Mai and Alquier (2017). We compare our adaptive MH implementation, denoted by “a-MH”, against the (random-walk) in Mai and Alquier (2017), denoted by “MH”; where all algorithms are run with 1000 iterations and 200 burnin steps. We run 50 independent samplers for each algorithm, and compute the mean of the square error (MSE),

$$\begin{aligned} \mathrm{MSE}:= \Vert {\hat{\rho }}-\rho \Vert _F^2 / d^2 \end{aligned}$$

for each method, together with their standard deviations. We also measure the mean absolute error of eigen values (MAEE) by

$$\begin{aligned} \mathrm{MAEE}:= \frac{1}{d} \sum _{i=1}^d | \lambda _i ({\hat{\rho }}) - \lambda _i (\rho ) |, \end{aligned}$$

where \( \lambda _i (A) \) are the eigen values of the matrix A.

5.2 Significantly speeding up

Fig. 1
figure 1

Plot to compare the running times (s) in log-scale for 10 steps of two algorithms in the setup of Setting 1, for the qubits \( n =2,4,6,7 \) (\( d= 4, 16, 64, 128 \)). “MH” is from reference Mai and Alquier (2017); “a-MH” is the Algorithm 1 without subsampling; “a-MH-30%” is the Algorithm 1 with subsampling 30%; “a-MH-60%” is the Algorithm 1 with subsampling 60%

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From Fig. 1, it is clear to see that our adaptive MH implementation is greatly faster than the previous implementation from Mai and Alquier (2017) by at least two orders of magnitude as the number of qubits increase. The data are simulated as in Setting 1 for \(n=2,4,6,7\) for which the dimensions of the density matrix are \( d= 4, 16, 64, 128 \) and of the empirical frequencies matrices \( [{\hat{p}}_{{\mathbf {a}}, {\mathbf {s}}}] \) are \( 9\times 4, 81 \times 16, 729\times 64, 2187\times 128 \). More specifically, for \(n = 6 \), our adaptive MH gives \(\sim \)115.9 times speedup comparing to the naive “MH” algorithm in Mai and Alquier (2017) and for \( n=7 \) the speedup is \(\sim \)251.1 times.

In addition, subsampling approaches also save the computational times respectively with the volume of the subsets, for example “a-MH-30%” will save the computational time by 2/3 while “a-MH-60%” will save the computational time by 1/3 of the full data approach “a-MH”. We note that these improvements are quite significant for practical quantum tomography where computational time is a precious resource.

5.3 Tuning parameters via acceptance rate

The tuning parameters \( \beta _y , \beta _z \) are chosen such that the acceptance rate of Algorithm 1 is between 0.15 and 0.3. This interval is chosen to enclose 0.234, the optimum acceptance probability for random-walk Metropolis-Hastings (under assumptions) (Gelman et al. 1997). For example, as in our experiments, for \( n=2 \) qubits: \( \beta _y = 0.33 , \beta _z = 0.2 \); for \( n=3 \) qubits: \( \beta _y = 0.03 , \beta _z = 0.03 \) and for \( n=4 \) qubits: \( \beta _y = 0.03 , \beta _z = 0.02 \) (all are run with \(\alpha =1, \lambda = m/2\)). We note that as the number of qubits n increase, these tuning parameters tend to be smaller and smaller to assure that the acceptance rate is between 0.15 and 0.3.

As an illustration, we conduct some simulations with \(n = 4 \) qubits in Setting 2. It can be seen from Fig. 2 that the acceptance rate between 0.2 and 0.3 would be optimal, as in Gelman et al. (1997). Where as high acceptance rate like 0.7 could make the algorithm be trapped at local points, and very small acceptance rate as 0.1 could make the algorithm converge slower.

Fig. 2
figure 2

Boxplots to examine the effect of the acceptance rate to MSE. The simulations are run within Setting 2 for \(n=4 \)

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Fig. 3
figure 3

Plots to compare the errors of two algorithms in different settings and with varying the number of qubits \( n =2,3,4, 5 \). The top 2 boxplots from left to right are in Setting 1, the bottom 2 boxplots from left to right are in Setting 2. “MH” is from reference Mai and Alquier (2017); “a-MH” is the Algorithm 1 with full data

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5.4 Similar accuracy with less variation

In term of accuracy performance, Fig. 3 compares the performance of our “a-MH” algorithm with the “MH” algorithm in various settings and varying the number of qubits \( n=2,3,4,5 \). The results show that both algorithms share similar accuracy in term of both considered errors (MSE and MAEE). However, it shows a clear improvement that our proposed adaptive algorithm yields much stable results (with less variation) compared to the naive MH approach as expected.

Results on subsampling are given in Fig. 4, where we further examine the performance of the “a-MH” algorithm with full data against subsampling the data by 60% and 30%. The outputs show that the subsampling approaches return comparable results. More specifically, in the cae of low-rankness (Setting 1), the subsampling approaches share similar accuracy (with higher variation) with the full data approach. In the case of mixed state (Setting 2), the subsampling approaches seem to return smaller error in term of mean squared errors, however their mean absolute error of eigen values (MAEE) are slightly higher than the full data approach. This can be explained as the target distribution of the subsampling algorithm is just an approximation of the target distribution in the full setting, thus the mean posterior could be well approximated but higher variation, (Quiroz et al. 2018b; Maire et al. 2019; Quiroz et al. 2018a; Alquier et al. 2016a).

Additional simulation regarding sensitivity analysis for different values of \( \lambda \) and \( \alpha \) are given in Fig. 5 in the Appendix.

Fig. 4
figure 4

Plots to compare the error of two algorithms in different settings and with \( n =4\). The top 2 boxplots from left to right are in Setting 1, the bottom 2 boxplots from left to right are in Setting 2. “adMH” is the Algorithm 1 with full data; “sub30adMH" is “adMH” with subsampling 30% the data; “sub60adMH" is “adMH” with subsampling 60% the data

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6 Discussion and conclusion

We have introduced an efficient sampling algorithm for Pseudo-Bayesian quantum tomography, especially for the prob-estimator. Our approach use a preconditioned proposal and subsampling Metropolis-Hasting implementation which shows a clear improvement in convergence, computation and computational time comparing with a naive MH implementation. We would like to mention that such an improvement is significantly important for practical quantum state tomography.

As suggested by one of the anonymous reviewer, in practice, one could change the tuning parameters \( \beta _y , \beta _z \) in Algorithm 1 dynamically adaptively. For example, one could change the values of these parameter every fixed steps (say 500 or 1000 steps) so that the acceptance rate is between 0.15 and 0.3. This could be an important step to obtain a better mixing rate in the chain.

Last but not least, faster algorithms based on optimization, such as Variational inference (Alquier et al. 2016b), for Bayesian quantum tomography would be an interesting research problem. However, it should be noted that the analysis of the uncertainty quantification when using Variational inference is not known up to present, while this matter is an important aspect in the problem of quantum state estimation.