Optimal experiment design for quantum state tomography

TL;DR

This paper introduces an integer programming approach to optimize measurement settings in quantum state tomography, reducing the number of measurements needed and simplifying experimental procedures.

Contribution

The authors develop a novel integer programming method to identify minimal measurement sets for quantum state tomography, improving efficiency over previous greedy strategies.

Findings

Significantly fewer measurement settings are required using the proposed method.

The approach is effective in practical NMR quantum systems.

It simplifies measurement schemes, reducing experimental effort.

Abstract

Quantum state tomography is an indispensable but costly part of many quantum experiments. Typically, it requires measurements to be carried in a number of different settings on a fixed experimental setup. The collected data is often informationally overcomplete, with the amount of information redundancy depending on the particular set of measurement settings chosen. This raises a question about how should one optimally take data so that the number of measurement settings necessary can be reduced. Here, we cast this problem in terms of integer programming. For a given experimental setup, standard integer programming algorithms allow us to find the minimum set of readout operations that can realize a target tomographic task. We apply the method to certain basic and practical state tomographic problems in nuclear magnetic resonance experimental systems. The results show that, considerably less readout operations can be found using our technique than it was by using the previous greedy search strategy. Therefore, our method could be helpful for simplifying measurement schemes so as to minimize the experimental effort.

Tables

Table 1. Table 1: Examples of optimal pulse set that yields complete tomography on n 𝑛 n -spin homonuclear systems for n 𝑛 n between 1 and 5.

$n$	$f_n^{\ast }$	A solution instance
1	2	$I$, $R_x$
2	4	$II$, $R_x^2$, $R_y^2$, $R_x^1{R}_x^2$
3	7	$III$, $R_y^3$, $R_y^1$, $R_y^2{R}_y^3$, $R_x^1{R}_y^2{R}_x^3$, $R_x^1{R}_x^2{R}_y^3$, $R_x^1{R}_x^2{R}_x^3$
4	15	$IIII$, $R_x^4$, $R_x^1{R}_x^4$, $R_x^1{R}_y^4$, $R_y^1{R}_y^2$, $R_x^2{R}_y^3{R}_y^4$, $R_y^2{R}_x^3{R}_y^4$,
		$R_y^2{R}_y^3{R}_y^4$, $R_x^1{R}_x^2{R}_x^3$, $R_x^1{R}_y^2{R}_y^3$, $R_y^1{R}_x^3{R}_x^4$, $R_y^1{R}_y^2{R}_y^3$,
		$R_x^1{R}_x^2{R}_x^3{R}_x^4$, $R_y^1{R}_x^2{R}_x^3{R}_x^4$, $R_y^1{R}_x^2{R}_y^3{R}_y^4$
5	33	$IIIII$, $R_x^5$, $R_x^4{R}_y^5$, $R_y^4{R}_y^5$, $R_x^3{R}_y^5$, $R_y^3{R}_y^5$, $R_x^2{R}_x^3{R}_x^4$,
		$R_y^2{R}_y^3{R}_y^4$, $R_x^1{R}_y^3{R}_x^4$, $R_x^1{R}_x^2{R}_y^4$, $R_x^1{R}_y^2{R}_y^5$, $R_x^1{R}_y^2{R}_x^3$,
		$R_y^1{R}_x^3{R}_y^4$, $R_y^1{R}_y^3{R}_y^5$, $R_y^1{R}_x^2{R}_y^5$, $R_y^1{R}_x^2{R}_y^3$, $R_y^1{R}_y^2{R}_y^5$,
		$R_y^1{R}_y^2{R}_x^4$, $R_x^2{R}_x^3{R}_x^4{R}_x^5$, $R_y^2{R}_y^3{R}_x^4{R}_x^5$, $R_y^2{R}_y^3{R}_y^4{R}_x^5$,
		$R_x^1{R}_y^3{R}_x^4{R}_x^5$, $R_x^1{R}_x^2{R}_y^4{R}_x^5$, $R_x^1{R}_y^2{R}_x^3{R}_x^5$, $R_y^1{R}_x^3{R}_y^4{R}_x^5$,
		$R_y^1{R}_x^2{R}_y^3{R}_x^5$, $R_y^1{R}_y^2{R}_x^4{R}_x^5$, $R_x^1{R}_x^2{R}_x^3{R}_x^4{R}_y^5$,
		$R_x^1{R}_x^2{R}_x^3{R}_y^4{R}_y^5$, $R_x^1{R}_x^2{R}_y^3{R}_x^4{R}_y^5$, $R_x^1{R}_x^2{R}_y^3{R}_y^4{R}_y^5$,
		$R_y^1{R}_y^2{R}_x^3{R}_x^4{R}_y^5$, $R_y^1{R}_y^2{R}_x^3{R}_y^4{R}_y^5$

Equations

$$X=\left(\begin{array}[]{cc}0&1\\ 1&0\end{array}\right),Y=\left(\begin{array}[]{cc}0&-i\\ i&0\end{array}\right),Z=\left(\begin{array}[]{cc}1&0\\ 0&-1\end{array}\right),$$

$$\min$$

$$s.t.$$

$$x_j\in \left\{0,1\right\}.$$

$$H=\sum _{k=1}^n{\Omega }_k{Z}_k/2+\pi \sum _{k<j}^n{J}_{kj}{Z}_k{Z}_j/2,$$

$$F(t)$$

$$=\mathrm{Tr}\left[\rho \cdot e^{iHt}\sum _m\left(X_m+i{Y}_m\right)e^{-iHt}\right].$$

$$\mathcal{O}=\left\{X,Y\right\}\otimes {\left\{I,Z\right\}}^{\otimes n-1}.$$

$$\mathcal{U}=\left\{V{W}_{1j}:V\in {\left\{I,R_x,R_y\right\}}^{\otimes n};j=1,...,n\right\}.$$

$${\mathcal{P}}^{(1)}$$

$${\mathcal{P}}^{(2)}$$

$${\mathcal{U}}^{(1)}=I\otimes {\left\{I,R_x,R_y\right\}}^{\otimes (n-1)};$$

$${\mathcal{U}}^{(2)}=\left\{V{W}_{1j}:V\in {\left\{I,R_x,R_y\right\}}^{\otimes n};j=2,...,n\right\}.$$

$$\mathbf{P}$$

$$\left\{I,Z\right\}\otimes \left\{X,Y\right\}\otimes {\left\{I,Z\right\}}^{\otimes n-2},$$

$$I\otimes \left\{V{W}_{2j}:V\in {\left\{I,R_x,R_y\right\}}^{\otimes n-1};j=2,...,n\right\}).$$

$$\mathcal{O}=\bigcup _{m=1}^n\left({\left\{I,Z\right\}}^{\otimes (m-1)}\otimes \left\{X_m,Y_m\right\}\otimes {\left\{I,Z\right\}}^{\otimes n-m}\right).$$

$$\mathcal{U}={\left\{I,R_x,R_y\right\}}^{\otimes n}.$$

Full text

I Introduction

The problem of estimating an unknown quantum state is of fundamental importance in quantum physics PR04 and especially in the field of quantum information processing, such as quantum computation NC00 , quantum cryptography GRTZ02 , and quantum system identification BY12 . Quantum state tomography aims to determine the full state of a quantum system via a series of quantum measurements. It has become an indispensable tool in almost any experimental physical setup. The standard tomography procedure applied for complete reconstruction of a $d$-dimensional quantum state consists in projecting the density operator with respect to at least $(d^{2}-1)$ measurement operators. A reconstruction based on Linear-squares inversion OWV97 or maximum-likelihood estimation TZERZ11 is then used to calculate the best-fit density matrix for the experimentally acquisited data set. Apparently, tomography is not an efficient process and can be extremely computationally costly for even modest-sized systems.

In recent years, state tomography has been an increasingly challenging task as the number of controllable qubits in quantum experiments is steadily growing. With the rapid progress of experimental control techniques, the size of quantum systems with entanglement or coherence prepared in the laboratory has already grown to 8-10 qubits in photonic systems Guo11 ; Pan12 ; Pan10 , 12 qubits in nuclear magnetic resonance (NMR) systems Ray06 and to even 14 qubits in ion traps Blatt11 . Needless to say, performing state estimation tasks on such systems is tedious and time-consuming. And improved techniques for quantum state tomography would certainly impact a wide range of applications in experimental physics. For example, Ref. H05 used hundreds of thousands of measurements and weeks of post-processing to get a maximum likelihood estimate of an entangled state of 8 trapped-ion qubits. Later, this experiment was simplified as there was put forward a much more economic tomographic scheme, which is based on the concept of mutually unbiased bases and promises to reduce about 95% of the number of measurements required KMFS08 . To give another example, in Ref. Guo16 it was shown that, in reconstructing a 14-qubit state, using both smart choice of the state representation and parallel graphic processing unit programming can speed up the post-processing by a factor of $10^{4}$. Besides of these technical improvements, there also exist various theoretical approaches that are devoted to enhance the capability of quantum state tomography GLFBE10 ; TWGKSW10 ; Cramer10 . Most of them either extract partial information or exploit some prior information about the state to be reconstructed.

In this paper, we are concerned with the design of the measurement scheme in a tomographic experiment. Our study is primarily motivated by a problem which is present in many experimental platforms, namely state tomography often involves informationally overcomplete measurements. The reason can be stated as follows. Normally, a tomographic experiment consists of a series of different measurement settings, and from each single measurement setting a bunch of data is recorded. Here, a measurement setting refers to a particular configuration of the experimental measurement apparatus. For example, in photonic systems one can tune the waveplates and polarizers to make arbitrary local polarization measurements, so a setting means the choice of one observable per qubit and repeated projective measurements in the observables’ eigenbases AJK05 . In NMR, a measurement setting corresponds to taking a spectrum. Because NMR experiments are performed on a large ensemble of molecules, the expectation values of the observables (not necessarily compatible) can be read out from a single spectrum VC05 . In both platforms, there could be considerable overlap between the experimental outcomes acquired from different measurement settings, that is, there are redundant measurements. Therefore, one would ask about what is the minimum number of measurement settings that suffice to determine the state of the system. A judicious experiment design would certainly help improve the efficiency of tomographic reconstruction. The purpose of this paper is to address this question via integer programming techniques. In the following, we shall first formulate the problem of optimal tomographic experiment design in terms of integer programming. And then, we concentrate on optimizing the design of readout pulse set in the context of nuclear magnetic resonance.

II Optimal Experiment Design

We restrict our consideration to the case of $n$ qubits—higher-dimensional systems can be treated similarly. An $n$-qubit system’s state is represented by a $2^{n}$-dimensional density matrix denoted as $\rho$, which is Hermitian, semipositive definite and has unit trace. A convenient and equivalent description of the quantum system is given by the Bloch vector. Let $\left\{B_{k}\right\}_{k=1}^{4^{n}-1}$ be some orthonormal basis for the space of traceless Hermitian operators satisfying that for any $k,j=1,...,n$, there is $\operatorname{Tr}(B_{k}B_{j})/2^{n}=\delta_{kj}$. Then decomposed with respect to this basis, $\rho$ can be viewed as a point $\bm{r}$ in a $(4^{n}-1)$-dimensional real vector space: $\rho=I^{\otimes n}/2^{n}+\sum_{k=1}^{4^{n}-1}{\bm{r}_{k}B_{k}}$ with $I$ being the $2$-dimensional identity matrix and $\bm{r}_{k}=\operatorname{Tr}(\rho B_{k})/2^{n}$. Clearly, full tomography amounts to measuring all of the quantities $\left\{\operatorname{Tr}(\rho B_{1}),...,\operatorname{Tr}(\rho B_{4^{n}-1})\right\}$.

Experimenters primarily work in two different bases, the computational basis and the product operator basis. In the computational basis the rows and columns of the density matrix $\rho$ are labeled by the binary expansion of their indices from $\left|0\cdots 0\right\rangle$ to $\left|1\cdots 1\right\rangle$. The product operator basis, defined as $\mathcal{P}_{n}=\left\{P_{k}\right\}_{k=1}^{4^{n}-1}=\left\{I,X,Y,Z\right\}^{\otimes n}/\left\{I^{\otimes n}\right\}$ where $X,Y,Z$ are the three Pauli matrices

[TABLE]

is a commonly used tool in describing pulse control experiments. It provides at the same time physical insight of the experimental setup (e.g., in NMR) and computational convenience SELBE84 . In the following we shall work on the product operator basis $\mathcal{P}$, but there is no problem in extending our analysis to other bases.

In performing a tomographic experiment, we send multiple copies of the state $\rho$ to our measurement apparatus. The apparatus can be configured in different settings. Suppose that under a specific measurement setting, we can read out the information for the following set of operators: $\mathcal{O}=\left\{O_{1},...,O_{k},...\right\}$, here $O_{k}\in\mathcal{P}$. The experimental tomography procedure employs a series of measurement settings, each corresponding to observation of a different set of operators. Here, the switch between measurement settings is implemented through either changing the configuration of the detectors or adding a unitary readout operation before data acquisition. This can be readily seen from the equality $\operatorname{Tr}(U\rho U^{\dagger}E)=\operatorname{Tr}(\rho UEU^{\dagger})$, where $E$ is an arbitrary observable and $U\in SU(2^{n})$. For instance, in order to measure the three Cartesian components of a spin, and if we can only observe $X$ and $Y$ in one experimental setting, then we will need two readout operations, which can be selected from the set $\left\{I,R_{x},R_{y}\right\}$, here $R_{x}$, $R_{y}$ is the $\pi/2$ rotation about $x$, $y$ axis respectively.

Now suppose we have the following experimentally available set of readout operations: $\mathcal{U}=\left\{U_{1},...,U_{j},...\right\}$. Denote $\mathcal{S}=\left\{S_{1},...,S_{j},...\right\}$ where $S_{j}$ corresponds to the set of measurement operators generated through $U_{j}$: $S_{j}=\left\{U_{j}O_{k}U_{j}^{\dagger}|O_{k}\in\mathcal{O}\right\}$. We shall assume that $S_{j}\subseteq\mathcal{P}$ for any $j$. Here is some abuse of notation as actually we should ignore global phase and coefficient. Then $\mathcal{S}$ is a collection of $\left|\mathcal{U}\right|$ subsets of $\mathcal{P}$, each containing $\left|\mathcal{O}\right|$ elements. Clearly, to ensure full state tomography, a necessary condition is that $\mathcal{P}$ should be covered by $\mathcal{S}$, that is, $\mathcal{P}=\bigcup_{j}S_{j}$. Now, we can state the central problem of this work, that is, to identify the smallest sub-collection of $\mathcal{S}$ whose union equals $\mathcal{P}$. More formally, we are considering a standard set cover problem, which we denote by $\mathbf{P}(\mathcal{P},\mathcal{O},\mathcal{U})$: given $\mathcal{P}$, $\mathcal{O}$, and $\mathcal{U}$, we want to select a subset of readout operations $\left\{U_{j}\right\}\subseteq\mathcal{U}$ with its number of elements as small as possible and such that $\mathcal{P}$ is covered by the set $\left\{U_{j}O_{k}U_{j}^{\dagger}|O_{k}\in\mathcal{O},\left\{U_{j}\right\}\subseteq\mathcal{U}\right\}$. Fig. 1 shows a simple instance of the problem.

Set cover problem is known to be NP-hard in general, meaning that to find an efficient algorithm that can solve it in reasonable time is unlikely. It is worthwhile to study heuristics for solving the problem with the goal of obtaining a performance guarantee 􏰏or approximation guarantee􏰐 on the heuristic􏰃. Practically, greedy strategy is widely used for set cover problems. Greedy algorithm proceeds according to a simple rule: in each step, choose the set $S_{j}$ containing the largest number of uncovered elements. The algorithm ends until all elements of $\mathcal{P}$ are covered. Ref. LM04 has exploited this method in designing readout pulse set in NMR. Note that greedy algorithm generally does not yield the optimal result. Different choices for the first few readout operations in the iteration yield lists that are slightly longer or shorter than those shown. It is in essence an approximation algorithm, which achieves an approximation ratio of $\Theta(\ln|\mathcal{P}|)$ GT15 . It can even be shown that no polynomial-time approximation algorithm can achieve a much better approximation bound Fei98 .

Here, we attempt to find the optimal solution for relatively small-sized quantum systems. To this end, we resort to integer linear programming formulation of the set cover problem Vazirani04 . Let $x$ denote a $|\mathcal{U}|$-element column vector, in which each element $x_{j}$ is a zero-one variable. The intention is that $x_{j}=1$ iff set $S_{j}$ is chosen in the optimal solution. Let $f(x)=\sum_{j}c_{j}x_{j}$ denote the cost function, where $c_{j}>0$ is the cost corresponding to the choice of $S_{j}$. In the case of minimum set cover problem, the cost function is just $f(x)=\sum_{j}x_{j}=\left\|x\right\|_{1}$. Let $A$ be a $(4^{n}-1)\times\left|\mathcal{U}\right|$ dimensional matrix with its entries given by $A_{kj}=1$ ($k=1,...,4^{n}-1$) if $P_{k}\in S_{j}$, and 0 otherwise. Now we have the following zero-one integer programming problem:

[TABLE]

There are a variety of algorithms that can be used to solve integer linear programs exactly, which we do not intend to expand here. Interested readers are referred to CBD09 ; Margot10 for the basics. For now, we make several discussions:

(1) Choose readout operations. For a quantum system that allows for universal control, readout operation $U$ can be chosen from the Clifford group $\mathcal{C}$. This is due to that the Clifford group is the normalizer of the Pauli group Gottesman97 , i.e., for any $U$ a Clifford operation and $O$ a Pauli observable, there is that if ignoring global phase and coefficient, $UOU^{\dagger}$ gives again a Pauli observable. However, the size of Clifford group is $|\mathcal{C}(n)|=8\cdot\prod_{k=1}^{n}2(4^{k}-1)4^{k}$, which makes the corresponding integer programming problem soon become too huge to handle. Therefore, it is difficult to consider the entire Clifford group in the integer programming approach. More realistically, we would restrict to just local operations. But notice that in many practical cases the set of single qubit rotations is not sufficient for covering the whole measurement basis, and then local Clifford operations should be considered.

(2) Cost function. Integer programming allows to consider different cost functions. In practice, more often than not, technical constraints permit only a nonideal set of measurements. For example, the qubits can be individually addressed whereas nonlocal quantities can not be measured directly. It is also very common that nonlocal operations are less accurate than local operations. In such cases, we would prefer to choose local operations, and this can be achieved simply through assigning low costs to these preferred operations.

(3) Symmetry Consideration. Many practical tomographic instances contain a great deal of symmetry. One direct consequence is that the optimal solution is not unique. A very useful technique that allows substantial reduction of the amount of computation required in running integer programming algorithms is to exploit the symmetry of the problem considered Ostrowski11 ; Margot10 . It is desirable to use professional softwares (e.g., Gurobi ) to run integer programming algorithms as they have taken into account of the symmetry issue and so can behave much faster.

III NMR Tomography

We first briefly describe the basics of state tomography experiments in NMR Lee02 . We consider weakly coupled liquid-state NMR systems. An NMR sample is placed in a strong static magnetic field. The direction of the static field is, by convention, defined as the $z$ axis. The system Hamiltonian takes the following form

[TABLE]

where $\Omega_{k}$ is the precession frequency of the $k$-th spin under the static field, and $J_{kj}$ is the coupling between the $k$th and $j$th spin. In NMR experiment, the sample is wounded with a detection coil. The precessing magnetization of the sample is detected by the coil and constitutes the free induction decay (FID). The induced signal is the sum of a number of oscillating waves of different frequencies, amplitudes, and phases. In the spectrometer, this is recorded using two orthogonal detection channels along the $x$- and $y$-axes, known as quadrature detection. The FID is then subjected to Fourier transformation, and the resulting spectral lines are fit, yielding a set of measurement data.

More precisely, let $\rho$ denote the state at the start of the sampling stage, then the measured time-domain signal $F(t)$ resulted from the rotating bulk magnetization is essentially a pair of ensemble averages:

[TABLE]

From the expression, the record of the FID signal $F(t)$ can be thought of as that the measured operators are $\mathcal{O}=e^{iHt}\sum_{k}\left(X_{k}+iY_{k}\right)e^{-iHt}$, which can be calculated easily when the Hamiltonian $H$ is specified. Because $H$ is composed of $Z$- and $ZZ$- terms, the measurement operator set $\mathcal{O}$ consists of only single-quantum coherence operators. When it is desired to measure other operators, readout pulses should be appended to the experiment.

Now, we give several examples showing how our method developed in the previous section applies to NMR state tomography.

(1) Tomography via a single probe qubit. In certain circumstances, we can do better than the numerical optimization: we can write down an analytic expression for an optimal readout scheme. Here, we develop a provably optimal scheme, where we intend to perform tomography via just a single probe qubit. Of course, to make the scheme work, two conditions must be assumed in advance: (i) the probe qubit is coupled to each of the other $n-1$ qubits; (ii) the multiplets corresponding to the probe qubit can be well resolved. As we have described, the set of measurement operators are just single-quantum coherences, so there is

[TABLE]

As such, we would transfer the information of other qubits to the probe qubit before detection. Let $W_{kj}$ denote the SWAP operation between qubit $k$ and $j$. Our candidate readout operations will be

[TABLE]

Let $f_{n}$ denote the number of experiments used. Now we show that the lower bound for the number of experiments necessary to constructing the density operator is

Proposition**.**

$f^{*}_{n}=(3^{n}+1)/2$.

Proof.

The correctness of the formula when $n=1$ is trivial. For $n>1$, the idea to the proof is to find a recursive relation.

First we show that $(3^{n}+1)/2$ is a lower bound. Notice that any $U\in\mathcal{U}$ can not change the weight of any Pauli element. So each experiment only gives observation results of two weight-$n$ Pauli elements. Since there are $3^{n}$ weight-$n$ Pauli elements in $\mathcal{P}$, so at least $(3^{n}+1)/2$ readout operations are needed.

Now we show that this lower bound can be achieved. We provide a constructive way of finding an optimal solution. Divide $\mathcal{P}$ into two parts: $\mathcal{P}=\mathcal{P}^{(1)}\cup\mathcal{P}^{(2)}$, where

[TABLE]

To cover $\mathcal{P}^{(1)}$, we select readout operations from

[TABLE]

to cover $\mathcal{P}^{(2)}$, we select readout operations from

[TABLE]

As $\mathcal{U}^{1}$ and $\mathcal{U}^{2}$ has no intersections, we get two seperate subproblems:

The subproblem $\mathbf{P}(\mathcal{P}^{(1)},\mathcal{O},\mathcal{U}^{(1)})$ is equivalent to problem $\mathbf{P}(\left\{I,X,Y,Z\right\}^{\otimes(n-1)},\left\{I,Z\right\}^{\otimes n-1},\left\{I,X,Y\right\}^{\otimes(n-1)})$. Since there are $3^{(n-1)}$ weight-$(n-1)$ Pauli elements and each experiment can only measure one of them, thus there must need $3^{(n-1)}$ experiments.

The subproblem $\mathbf{P}(\mathcal{P}^{(2)},\mathcal{O},\mathcal{U}^{(2)})$ can be reduced to

[TABLE]

That is, any solution to the latter problem can be mapped to a solution to the former by simply changing the probe qubit from 2 to 1. Note that the latter problem is essentially $\mathbf{P}(\mathcal{P}(n-1),\mathcal{O}(n-1),\mathcal{U}(n-1))$. Therefore, our construction has a recursive relation: $f_{n}=f_{n-1}+3^{n-1}$. Starting with $f^{*}_{1}=2$, we then get $f_{n}=(3^{n}+1)/2$. Together with the bound analysis at the beginning, we conclude that the lower bound $(3^{n}+1)/2$ is exact. ∎

Example 1**.**

Iodotrifluroethylene (C2F3I) dissolved in d-chloroform. This molecule was used as a quantum information processor in Ref. Luo16 ; Li16 . See the following figure for the sample’s molecular structure:

C1CF2F1IF3

Here, the 13C (C1) nucleus and the three 19F nuclei (F1, F2 and F3) constitute a four-qubit system. For this molecule, only C1 can be adequately observed. Therefore to observe those operators that are relevant to the state of the fluorines it is necessary to swap the state between them and the carbon before observation. So readout operations can be chosen from the set in Eq. (3). According to our proposition, the minimum number of readout single-qubit rotation pulses is 41.

(2) Now we consider homonuclear systems, assuming that all peaks on the spectrum can be adequately resolved and observed. In homonuclear systems, all the multiplets are experimentally observed on the same spectrometer channel, and so one readout operation, namely one spectrum, contains $n$ well-resolved multiplets and yields $n2^{n}$ expansion coefficients. In such an ideal case, we can observe all the single-quantum transition operators from the experimental spectrum. In other words, we will have the following collection of Pauli measurements that can be accessed in a single experiment

[TABLE]

One can choose the following readout operation set

[TABLE]

Ref. LM04 referred $\mathcal{U}$ as the canonical tomographic pulse set, and employed greedy algorithm to search for smaller readout pulse set. It turns out that, unlike the previous example, it is hard to analytically construct an optimal scheme for the current problem. Here we resort to integer programming. Fig. 2 shows our running results, which are listed together with the methods of canonical tomography and greedy search. From the figure, we can see that for a 6 qubit-system, using the measurement scheme found by integer programming would save around 20% of the experiment time compared with that if we use greedy method. Moreover, integer programming allows to confirm that the obtained solution is indeed optimal for system’s size up to 5; see Table 1. These results clearly demonstrate the usefulness of the integer programming technique, that we could get appreciable improvements over what has been obtained before.

IV Summary

Quantum state tomography plays an essential role in many quantum information processing experiments. Developing techniques that allow simplification of state tomography experiments is particularly pressing in regard of the situation that ever larger sized quantum devices are emerging in laboratories. In this work, we have studied the application of integer programming to the problem of reducing the number of required measurement settings and the computational complexity of data processing. The presented test examples confirm the usefulness of the integer programming approach. Our method can be easily incorporated into other existing tomographic strategies Jackson13 ; NSPW10 ; Ma16 . Also, it is straightforward to generalize our results to quantum process tomography experiments. It is the hope that integer programming formulation, as developed in this work, will become a useful tool in future tomographic experiments for increasingly large quantum systems, overcoming the roadblock against further development in quantum technologies.

Acknowledgments. Jun Li is supported by the National Basic Research Program of China (Grants No. 2014CB921403, No. 2016YFA0301201, No. 2014CB848700 and No. 2013CB921800), National Natural Science Foundation of China (Grants No. 11421063, No. 11534002, No. 11375167 and No. 11605005), the National Science Fund for Distinguished Young Scholars (Grant No. 11425523), and NSAF (Grant No. U1530401). Bei Zeng is supported by NSERC and CIFAR.