O ne of the essential tasks in quantum technology is to verify the integrity of a quantum state1. Quantum

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1 OPE SUBJECT AREAS: QUATUM IFORMATIO QUATUM MECHAICS Receive 30 September 2013 Accepte 28 ovember 2013 Publishe 13 December 2013 Corresponence an requests for materials shoul be aresse to G.Y.X. eu.cn) Quantum State Tomography via Linear Regression Estimation Bo Qi 1, Zhibo Hou 2,LiLi 2, Daoyi Dong 3, Guoyong Xiang 2 & Guangcan Guo 2 1 Key Laboratory of Systems an Control, ISS, an ational Center for Mathematics an Interisciplinary Sciences, Acaemy of Mathematics an Systems Science, CAS, Beijing , P. R. China, 2 Key Laboratory of Quantum Information, University of Science an Technology of China, CAS, Hefei , P. R. China, 3 School of Engineering an Information Technology, University of ew South Wales at the Australian Defence Force Acaemy, Canberra, ACT 2600, Australia. A simple yet efficient state reconstruction algorithm of linear regression estimation (LRE) is presente for quantum state tomography. In this metho, quantum state reconstruction is converte into a parameter estimation problem of a linear regression moel an the least-squares metho is employe to estimate the unknown parameters. An asymptotic mean square error (MSE) upper boun for all possible states to be estimate is given analytically, which epens explicitly upon the involve measurement bases. This analytical MSE upper boun can guie one to choose optimal measurement sets. The computational complexity of LRE is O( 4 ) where is the imension of the quantum state. umerical examples show that LRE is much faster than maximum-likelihoo estimation for quantum state tomography. O ne of the essential tasks in quantum technology is to verify the integrity of a quantum state1. Quantum state tomography has become a stanar technology for inferring the state of a quantum system through appropriate measurements an estimation 2 8. To reconstruct a quantum state, one may first perform measurements on a collection of ientically prepare copies of a quantum system (ata collection) an then infer the quantum state from these measurement outcomes using appropriate estimation algorithms (ata analysis). Measurement on a quantum system generally gives a probabilistic result an an iniviual measurement outcome only provies limite information on the state of the system, even when an ieal measurement evice is use. In principle, an infinite number of measurements are require to etermine a quantum state precisely. However, practical quantum state tomography consists of only finite measurements an appropriate estimation algorithms. Hence, the choice of optimal measurement sets an the esign of efficient state reconstruction algorithms are two critical issues in quantum state tomography. Many results have been presente for choosing optimal measurement sets to increase the estimation accuracy an efficiency in quantum state tomography Several soun choices that can provie excellent performance for tomography are, for instance, tetraheron measurement bases, cube measurement sets, an mutually unbiase bases 11. However, for most existing results, the optimality of a given measurement set is only verifie through numerical results 11. There are few methos that can analytically give an estimation error boun 12 14, which is essential to evaluate the optimality of a measurement set an the appropriateness of an estimation metho. For estimation algorithms, several useful methos incluing maximum-likelihoo estimation (MLE) 2,18 21, Bayesian mean estimation (BME) 2,22,23 an least-squares (LS) inversion 24 have been propose for quantum state reconstruction. The MLE metho simply chooses the state estimate that gives the observe results with the highest probability. This metho is asymptotically optimal in the sense that the estimation error can asymptotically achieve the Cramér-Rao boun. However, MLE usually involves solving a large number of nonlinear equations where their solutions are notoriously ifficult to obtain an often not unique. Recently, an efficient metho has been propose for computing the maximum-likelihoo quantum state from measurements with aitive Gaussian noise, but this metho is not general 21. Compare to MLE, BME can always give a unique state estimate, since it constructs a state from an integral averaging over all possible quantum states with proper weights. The high computational complexity of this metho significantly limits its application. The LS inversion metho can be applie when measurable quantities exist that are linearly relate to all ensity matrix elements of the quantum state being reconstructe 24. However, the estimation result may be a nonphysical state an the mean square error (MSE) boun of the estimate cannot be etermine analytically. Here, we present a new linear regression estimation (LRE) metho for quantum state tomography that can ientify optimal measurement sets an reconstruct a quantum state efficiently. We first convert the quantum state reconstruction into a parameter estimation problem of a linear regression moel 25. ext, we employ an LS SCIETIFIC REPORTS 3 : 3496 DOI: /srep

2 algorithm to estimate the unknown parameters. The positivity of the reconstructe state can be guarantee by an aitional least-squares minimization problem. The total computational complexity is O( 4 ) where is the imension of the quantum state. In orer to evaluate the performance of a chosen measurement set, an MSE upper boun for all possible states to be estimate is given analytically. This MSE upper boun epens explicitly upon the involve measurement bases, an can guie us to choose the optimal measurement set. The efficiency of the metho is emonstrate by examples on qubit systems. Results Linear regression moel. We first convert the quantum state tomography problem into a parameter estimation problem of a linear regression moel. Suppose the imension of the Hilbert space H of the system of interest is, an fv i g 2 {1 i~0 is a complete basis set of orthonormal operators on the corresponing Liouville space, namely, Tr V { i V j ~ ij, where { enotes the Hermitian ajoint an ij is the Kronecker function. Without loss of generality, let V i ~V { i an V 0 ~ ð1=þ 1 2 I, such that the other bases are traceless. That is Tr(V i ) 5 0, for, 2,, 2 {1. The quantum state r to be reconstructe may be parameterize as r~ I z X2 {1 H i V i, ð1þ where H i 5 Tr(rV i ). Given a set of measurement bases, each jyææyj(n) can be parameterize uner the bases f g 2 {1 V i i~0 as jyihy ð Þ ~ I z X2 {1 j n y ðnþ i V i, ð2þ where y ðnþ i ~Tr V i. When one performs measurements with measurement set on a collection of ientically prepare copies of a quantum system (with state r), the probability to obtain the result of jyææyj (n) is p n ~Tr r ~ 1 z X2 {1 H i y ðnþ i ¼ D 1 zh> : Assume that the total number of experiments is an /M experiments are performe on /M ientically prepare copies of a quantum system for each measurement basis jyææyj (n). Denote the corresponing outcomes as x ðnþ 1,, x(n), which are inepenent an ientically istribute. Let ^p n ~ x ð n Þ 1 z zx ðnþ an e n ~^p n {p n. Accoring to the central limit theorem 26, e n converges in istribution to a normal istribution with mean 0 an variance p n {p 2 n. Using (3), we have the linear regression equations for, 2,, M, ^p n ~ 1 z Hze n, where i enotes the matrix transpose. ote that ^p n, an Y (n) are all available, while e n may be consiere as the observation noise whose variance is asymptotically p n {p 2 n. Hence, the problem of quantum state tomography is ð3þ ð4þ converte into the estimation of the unknown vector H. Denote Y~ ^p 1 { 1,, ^p M { 1 > >,, X~ Y ðþ 1,,Y ðmþ e~ ð e1,, e M Þ >. We can transform the linear regression equations (4) into a compact form Y~XHze: ð5þ We efine the MSE as ETrð^r{r Þ 2, where ^r is an estimate of the quantum state r base on the measurement outcomes an E(?) enotes the expectation on all possible measurement outcomes. For a fixe tomography metho, ETrð^r{r Þ 2 epens on the state r to be reconstructe an the chosen measurement bases. From a practical viewpoint, the optimality of a chosen set of measurement bases may rely upon prior information but shoul not epen on any specific unknown quantum state to be reconstructe. In this paper, no prior assumption is mae on the state r to be reconstructe. Given a fixe tomography metho, we use the maximum MSE for all possible states (i.e., sup r ETrð^r{r Þ 2 ) as the inex to evaluate the performance of a chosen set of measurement bases. Linear regression estimation. To give an estimate with high level of accuracy an low computational complexity, we employ the LS metho, where the basic iea is to fin an estimate ^H LS such that ^H LS ~ argmin ^H Y{X ^H >W Y{X ^H, ð6þ where ^H is an estimate of H, an W is a iagonal weighting matrix. Since the objective function is quaratic, one has the LS solution as follows: ^H LS ~ X > {1X WX > WY: ð7þ The LS solution (7) can be calculate in a recursive way (see the Methos section). In practical experiments, the cost of time can be greatly reuce by employing a recursive reconstruction protocol since the estimate can be calculate recursively base on available ata at the same time of performing measurements to acquire ata. ote that if p n 5 1, we have alreay reconstructe the state as jyææyj (n) ;ifp n 5 0, we shoul choose the following measurement basis from the orthogonal complementary space of jyææyj (n). Hence, in general the smaller the variance of e n is, the more the information can be extracte by jyææyj (n). Therefore, the corresponing weight of the n-th regression equation shoul be bigger. It can be verifie that if all p n are known, the LS soution ^H LS satisfying ^H LS ~ argmin Y{X ^H >V Y{X ^H is asymptotically the minimum ^H variance unbiase estimator of H, where V is the inverse of iag p 1 {p 2 1,, p M{p 2 M. Hence, an appropriate choice of W is the inverse of iag ^p 1 {^p 2 1,, ^p M {^p2 M. However, for simplicity we consier the case where W 5 I, an the corresponing LS solution is ^H LS ~ X > X {1X > Y~ X > {1 X M X ^p n { 1, ð8þ where X > X~ X M. If the measurement bases are informationally complete or overcomplete, X T X is invertible. Using (5), (8) an the statistical property of the observation noise fe n g M (inepenent an asymptotically Gaussian), the estimate ^H LS has the following properties for a fixe set of chosen measurement bases: SCIETIFIC REPORTS 3 : 3496 DOI: /srep

3 1. ^HLS is asymptotically unbiase; 2. The MSE E ^H > LS {H ^HLS {H of ^H LS is asymptotically M h Tr {1X X> X > PX X > i {1 X, ð9þ where P~iag p 1 {p 2 1,, p M{p 2 M. ote that p n epens upon the state to be reconstructe an the measurement basis jyææyj (n) for,, M. Recall that the optimality of a chosen set of measurement bases shoul not epen upon any specific unknown quantum state to be reconstructe. We can take the supremum of equation (9) uner all possible states to get the performance inex for any given set of measurement bases as M 4 Tr {1~ M X> X 4 Tr X M {1. Positivity an computational complexity. Base on the solution ^H LS obtaine from (8), we can obtain a Hermitian matrix ^m with Tr^m~1 using (1). However, ^m may have negative eigenvalues an be nonphysical ue to the ranomness of measurement results. In this sense, ^m is calle pseuo linear regression estimation (PLRE) of state r. A goo metho of pulling ^m back to a physical state can reuce the MSE. In this paper, the physical estimate ^r is chosen to be the closest ensity matrix to ^m uner the matrix 2-norm. In stanar state reconstruction algorithms, this task is computationally intensive 21. However, we can employ the fast algorithm in 21 with computational complexity O( 3 ) to solve this problem since we have obtaine a Hermitian estimate ^m with Tr^m~1. Since an informationally complete measurement set requires M being O(2 ), the computational complexity of (1) an X T Y in (8) is O( 4 ). Although the computational complexity of calculating (X T X) 21 is generally O( 6 ), (X T X) 21 can be compute off-line before the experiment once the measurement set is etermine. Hence, the total computational complexity of LRE after the ata have been collecte is O( 4 ). It is worth pointing out that for n-qubit systems, X > X~ P M is iagonal for many preferre measurement sets such as tetraheron an cube measurement sets. Fig. 1 compares the run time of our algorithm with that of a traitional MLE algorithm. Since the maximum MSE coul reach 2 for the worst estimate, it is clear that our state reconstruction algorithm LRE is much more efficient than MLE with a small amount of accuracy sacrifice. Optimality of measurement bases. One of the avantages of LRE is that the MSE upper boun can be given analytically as M 4 Tr X M {1, which is epenant explicitly upon the measurement bases. ote that if the PLRE ^m is a physical state, then the MSE upper boun is asymptotically tight for the evaluation of the performance of a fixe set of measurement bases. Hence, to choose an optimal set, one can solve the following optimization problem:! X M {1 Minimize Tr ð10þ s:t: ~ {1, for,, M: The optimization problem can be solve in an off-line way by employing appropriate algorithms though it may be computationally intensive. We will iscuss this problem in other work. With the help of the analytical MSE upper boun, we can ascertain which one is optimal among the available measurement sets. This is emonstrate when we prove the optimality of several typical sets of measurement bases for 2-qubit systems. For 2-qubit systems, it is convenient to chose V i ~ p 1 ffiffi s l p ffiffi s m, where i 5 4l 1 m; l, m 5 0, 1, 2, 3; s 0 5 I 232, s 1 ~ 2 0 1, s ~ 0 {i, s i 0 3 ~ {1 If the form of the measurement bases is not restricte, the minimum of the MSE upper boun M 4 Tr {1 X> X for all possible measurement bases is 75. This minimum can be reache by using the Secons Time of LRE MSE of LRE Time of MLE MSE of MLE MSE n Figure 1 The run time an MSE of LRE an MLE for ranom n-qubit pure states mixe with the ientity 21. The realization of MLE use the iterative metho in 2. The measurement bases are from the n-qubit cube measurement set an the resource is n. The simulate measurement results for every basis YæÆY (i) are generate from a binomial istribution with probability p i 5 Tr( YæÆY (i) r) an trials /M. LRE is much more efficient than MLE with a small amount of accuracy sacrifice since the maximum MSE coul reach 2 for the worst estimate. All timings were performe in MATLAB on the computer with 4 cores of 3 GHz Intel i CPUs. SCIETIFIC REPORTS 3 : 3496 DOI: /srep

4 mutually unbiase measurement bases. While as in many practical experiments, if only local measurements can be performe, the minimum of the MSE upper boun M 4 Tr {1 99 X> X is. This minimum can be reache by using the 2-qubit cube or tetraheron measurement set. Fig. 2 shows the epenant relationships of the MSEs for Werner states on q (varying from 0 to 1) an ifferent number of copies using the cube measurement bases 9. The fact that the MSE of PLRE is larger than that of LRE emonstrates that the process of pulling ^m back to a physical state further reuces the estimation error. Discussion In the LRE metho, ata collection is achieve by performing measurements on quantum systems with given measurement bases. This process can also be accomplishe by consiering the evolution of quantum systems with fewer measurement bases. For example, suppose only one observable s is given, an the system evolves accoring to a unitary group {U t }. At a given time t, hs t i~tr U { ðþsu t ðþr t ~Tr ð st rþ: ð11þ Suppose one measures the observable s at time tt~1, ð, MÞon m ientically prepare copies of a quantum system. Denote the obtaine outcomes as s t 1,, st m, an their algebraic average as s t ~ st 1 z zst m. ote that s t 1 m,, st m are inepenent an ientically istribute. Accoring to the central limit theorem 26, e t ~s t { hs t iconverges in istribution to a normal istribution with s 2 t { h st i 2 mean 0 an variance. We have the following linear m regression equations s t ~Trðs t rþze t, t~1,, M, ð12þ which are similar to (4). Hence, we can use the propose LRE metho to accomplish quantum state tomography. The LRE metho can also be extene to reconstruct quantum states with a prior information 12,27 29 or states of open quantum systems. Actually, LRE can be applie whenever there are measurable quantities that are linearly relate to all ensity matrix elements of the quantum system uner consieration. In conclusion, an efficient state reconstruction algorithm of linear regression estimation has been presente for quantum state tomography. The computational complexity of LRE is O( 4 ), which is much lower than that of MLE an BME. We have analytically provie an MSE upper boun for all possible states to be estimate, which explicitly epens upon the use measurement bases. This analytical upper boun can assist to ientify optimal measurement sets. The LRE metho has potential for wie applications in real experiments. Figure 2 Mean square error (MSE) for Werner states (r q ~qjy { ihy { jz 1{q I with jy{ i~ jhv i{ VH p ffiffi j i ) with q (varying from 0 to 1) an 4 2 ifferent numbers of copies. The cube measurement set is use, where the MSE upper boun is 99. It can be seen that the MSE of PLRE is almost unchange for q g [0, 1], an is larger than the MSE of LRE. SCIETIFIC REPORTS 3 : 3496 DOI: /srep

5 Methos The recursive LS algorithm. For,, M, efine ^H n as X ^H n n ~ argmin W ii ^p i { 1 2 {Y ðþ> i ^H, ð13þ ^H where W ii is the i-th element of the iagonal of W, an ^H is an estimate of H. Hence, the LS solution ^H LS is equal to ^H M. From (7), we have Define Q n ~ ^H n ~ Xn Xn! " {1 X n W ii Y ðþ i Y ðþ> i Y ðþ i W ii ^p i { 1 # : ð14þ! {1 W ii Y ðþ i Y ðþ> i, a n ~ 1 {1 z Q n{1 : ð15þ W nn Using the matrix inversion formula (see, e.g., page 19 of 30 ) ða{bcdþ {1 ~A {1 za {1 BC {1 {DA {1 {1DA B {1, we have Q n ~Q n{1 {a n Q n{1 Q n{1 : From (14), (15) an (16), the recursive form of ^H n can be obtaine as ^H n ~ ^H n{1 za n Q n{1 ^p n { 1 {Y n ð Þ> ^Hn{1 ð16þ : ð17þ ote that Q n is not always invertible, especially when n is small. In orer to apply the recursive algorithm in this case, one may choose the initial value in (16) Q 0 being a given positive matrix, while ^H 0 being a given vector. From (16) an the matrix inverse formula, one has Q n ~ Xn! {1 W ii Y ðþ i Y ðþ> i zq {1 0 : Hence, the recursive LS algorithm can still be applie. Although the solution obtaine from (17) may be slightly ifferent from the solution obtaine using (14), this oes not affect the asymptotic properties of the LS solution. The minimum of the MSE upper boun. The MSE upper boun of 2-qubit states is M 4 Tr {1~ M X> X 4 Tr X M! {1 : ð18þ Minimizing this MSE upper boun is equivalent to minimizing Tr(X T X) 21. Denote the eigenvalues of X T X as l 1 l 2 l 15. Since for all possible measurement bases, we have y ðnþ 0 ~ 1 2, X 15 y ðnþ2 i~0 i ~1 for,, M, the problem is converte into the following conitional extremum problem: Minimize s:t: X 15 X 15 1, l i l i ~ 3 4 M: ð19þ It can be proven that X 15 1 reaches its minimum 300 l i M when l 1~ ~l 15 ~ M 20. Hence, the minimum of the MSE upper boun M 4 Tr {1 X> X for all possible measurement bases is 75 : It can be verifie that this minimum MSE upper boun can be reache by using the mutually unbiase measurement bases. If only local measurements can be performe, i.e., ~ jyihyj ðn,1þ 6jYihYj ðn,2þ,,, M, where jyææyj (n,1) an jyææyj (n,2) can be parameterize as jyihyj ðn,kþ ~ P 3 l~0 y ðn,kþ s l l p ffiffi, k 5 1, 2. An we 2 have y ðnþ i ~y ðn,1þ l y ðn,2þ m, where i 5 4l 1 m. Due to aitional constraints y ðn,kþ 0 ~ p 1 ffiffi, X 3 2 y ðn,kþ2 l~0 l ~1, for k 5 1, 2, an,, M, the problem of minimizing the MSE upper boun can be converte into the following problem: Minimize s:t: X 15 1, l i ðþ i X3 l i 1 M, 4 ð ii ÞX6 l i 1 2 M, ðiiiþ X15 l i ~ 3 4 M: ð20þ It can be proven that X 15 1 reaches its minimum 396 l i M when l 1~ ~l 6 ~ M 12, l 7 ~ ~l 15 ~ M 36. Hence, the minimum of the MSE upper boun M 4 Tr {1 X> X is 99. 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6 30. Horn, R. A. & Johnson, C. R. Matrix Analysis (Cambrige University Press, Cambrige, 1985). Acknowlegments The authors woul like to thank Lei Guo, Huangjun Zhu an Chuanfeng Li for helpful iscussion. The work in USTC is supporte by ational Funamental Research Program (Grants o. 2011CBA00200 an o. 2011CB ), ational atural Science Founation of China (Grants o an o ), Anhui Provincial atural Science Founation(o QA08). B.Q. acknowleges the support of ational atural Science Founation of China (Grants o , o , o an o ). D.D. is supporte by the Australian Research Council (DP ). Author contributions B.Q., L.L. an D.D. evelope the scheme base on linear regression moel, Z.-B.H., G.-Y.X. an G.-C.G. performe the numerical simulations. All authors iscusse the results an contribute to the writing of the paper. G.-Y.X. supervise the project. Aitional information Competing financial interests: The authors eclare no competing financial interests. How to cite this article: Qi, B. et al. Quantum State Tomography via Linear Regression Estimation. Sci. Rep. 3, 3496; DOI: /srep03496 (2013). This work is license uner a Creative Commons Attribution 3.0 Unporte license. To view a copy of this license, visit SCIETIFIC REPORTS 3 : 3496 DOI: /srep