Correcting detection error in quantum computation and state engineering through data processing

Quantum error correction in general is experimentally challenging as it requires significant expansion of the size of quantum circuits and accurate performance of quantum gates to fulfill the error threshold requirement. Here we propose a method much simpler for experimental implementation to correct arbitrary detection errors. The method is based on processing of data from repetitive experiments and can correct detection error of any magnitude, as long as the error magnitude is calibrated. The method is illustrated with its application to detection of multipartite entanglement from quantum state engineering.

Quantum error correction in general is experimentally challenging as it requires significant expansion of the size of quantum circuits and accurate performance of quantum gates to fulfill the error threshold requirement. Here we propose a method much simpler for experimental implementation to correct arbitrary detection errors. The method is based on processing of data from repetitive experiments and can correct detection error of any magnitude, as long as the error magnitude is calibrated. The method is illustrated with its application to detection of multipartite entanglement from quantum state engineering. A celebrated achievement in quantum information science is establishment of the error threshold theorem, which assures that any experimental error below a certain threshold value can be corrected in quantum computation or communication [1,2]. This theorem is important as it indicates there is no fundamental obstacle to realization of quantum computation with imperfect experimental devices, as long as the imperfection is small. The experimental realization of fault-tolerant quantum error correction, unfortunately, is still challenging. When all the experiment devices are subject to errors as it is the case for real applications, realization of fault-tolerance requires significant expansion of the size of quantum circuits for complicated encoding and accurate performance of quantum gates to fulfill the error threshold requirement, which is still beyond the capability of current experimental technology.
In this paper, we propose an alternative method to correct a special but practically important type of error -the detection error -in quantum information processing. The detection error is a common source of noise in many quantum information experiments. In particular, it is a significant obstacle to observation of multipartite entanglement in quantum state engineering [3]. We show here that this type of error can be corrected at any magnitude as long as the error magnitude has been calibrated (for instance, through prior test experiments). The detection error distorts the experimental data by a transformation that depends on the magnitudes of various error possibilities. When the relevant error magnitudes have been calibrated for the detectors by the prior test experiments, the form of the distortion transformation induced by the detection error is known, and then we can find a way to inverse this transformation to reconstruct the original signal. In this way, we can use imperfect detectors to simulate perfect detectors as long as their imperfection has been calibrated. The proposed method is straightforward for experimental implementation as it is based on data processing and removes the difficulty associated with fault-tolerant quantum encod-ing. To correct the detection error, we only require to repeat the same experiments by some additional rounds to have small statistical error for the inverse transformation. To illustrate its applications, the method is used to significantly improve the detection of multi-qubit entanglement and spin squeezing. In many cases, the signal of multipartite entanglement only becomes visible after the proposed correction of the detection error, in particular when the number of qubits is large.
Any measurements in quantum information can be reduced to population measurements in certain bases (including possibly several complementary bases). If we want to measure properties associated with a state ρ (generally mixed) of n qubits, in each chosen measurement basis, there are 2 n possible measurement outcomes. By measurements we determine the probability f i associated with each outcome i (i = 1, 2, · · · , 2 n ). For instance, if we repeat the same experiment N times and get the ith outcome N i times, we estimate the probability f i by f i = N i /N and its standard deviation (the error bar) by ∆f i = f i (1 − f i )/N using the binomial distribution. If the detectors are perfect, the measured probabilities f i just give the distribution g i ≡ i| ρ |i of the state ρ in the measurement basis {|i }. In reality, however, the detectors always have errors, which distort the distribution g i , making the measured distribution f i significantly different from g i . The purpose of this paper is to show how to reconstruct the real distribution g i from the measured distorted probabilities f i .
We first consider the case where the measurements haven individual addressing, and each qubit is measured by an independent detector. For detection on a qubit, the most general error model is characterized by a 2 × 2 matrix where p 0 (p 1 ) denotes respectively the error probability that the detector gives outcome 1 (0) for the input signal of 0 (1). For simplicity of notation, we assume the error matrix D has the same form for detection of each qubit (it is straightforward to generalize the formalism to the case where the error rates p 0 and p 1 in the D matrix are qubit-dependent). Furthermore, we assume p 0 and p 1 have been well calibrated by a prior test experiment. For instance, we may input a known state to the detector and can calibrate p 0 and p 1 easily from the measurement data.
For n qubits, the error model for the detection is then characterized by a 2 n × 2 n matrix M = [M ji ], with the element M ji corresponding to the probability of recording the outcome j with the input signal i. Assume the detection error rates on different qubits are independent to each other and the binary string i has n 0 zeros and n 1 = n − n 0 ones. If we need α flips from 0 to 1 and β flips from 1 to 0 to change the string from i to j, the matrix element M ji is given by The measured probabilities f j are connected with the real distribution g i through the distortion transformation To reconstruct the real signal g i from the measured distribution f j , in principle we only need to inverse the matrix M . However, as M is a huge 2 n × 2 n matrix, it is not clear how to inverse this matrix (it is even a question whether the inverse exists).
Our key observation is that the matrix M , with the elements given by Eq. (2), has a simple tensor product structure. It is straightforward to show by mathematical induction that where all the D k are identical and given by D in Eq. (1). Therefore, the inverse can be easily done in an analytic form with where the parameters p ′ 0 and p ′ 1 are given by Note that with the substitution in Eq. (5), M −1 and M have the same form except that p ′ 0 and p ′ 1 can not be interpreted as error rates any more since in general they are not in the range [0, 1]. The formula also shows that the inverse transformation M −1 always exists except for the special case with p 0 + p 1 = 1.
In some experimental systems we do not have the ability to resolve individual qubits. Instead, we perform collective measurements on n qubits by detecting how many qubits (denoted by j, j = 0, 1, · · · , n) are in the state |1 in a chosen detection basis (this is equivalent to measurement of the collective spin operator along a certain direction). In this case, the detection only has n + 1 outcomes for an n-qubit system. For collective measurements on n qubits, the detection error matrix is represented by an (n + 1) × (n + 1) matrix L = [L ij ]. The matrix element L ij corresponds to the probability to register outcome i when j qubits are in the |1 state. If the detection error matrix for an individual qubit is still given by D in Eq.
(1), we can directly calculate L ij from D: from signal j to i, if n 10 qubits flip from 0 to 1 and n 01 qubits flip from 1 to 0, with the constraints 0 ≤ n 01 ≤ j, 0 ≤ n 10 ≤ n − j and n 01 − n 10 = j − i, L ij is given by where B(n, p, k) ≡ n k p k (1 − p) n−k and we have let q = i − n 10 , and hence q satisfies the constraint max{0, i + j − n} ≤ q ≤ min{i, j}. As the dimension of the L matrix depends linearly on the qubit number n, it is typically not difficult to numerically calculate its inverse matrix L −1 if n is not very large. As will be shown in the appendix, there is also a simple analytic formula for L −1 = L −1 ij : if we denote the dependence of L ij in Eq. (6) on p 0 , p 1 as L ij = L ij (p 0 , p 1 ), we have where p ′ 0 , p ′ 1 are given by the simple substitution in Eq. (5). With the inverse matrix L −1 , the real signal g i can be similarly reconstructed from the measured data f j as g i = j L −1 ij f j . The above formulation can be extended straightforwardly to qudit (d-dimensional) systems where the individual detection error matrix D in Eq. (1) is replaced by a d×d matrix. For independent detection of n-qudits, the overall error matrix M still has the tensor-product structure as shown by Eq. (3), which allows easy calculation of M −1 from D −1 .
With the inverse error matrix M −1 , it is straightforward to reconstruct the true distribution g i from the measured data f i . The price we need to pay is that compared with ∆f i = f i (1 − f i )/N , there is an increase of the standard deviation (error bar) ∆g i in our estimate of g i by the formula g i = 2 n j=1 M −1 ij f j . With some tedious but straightforward calculation, we find As M −1 = n k=1 D −1 k and D −1 k has matrix element 1 − p ′ 0 ≈ e p > 1 (when p 0 ∼ p 1 ∼ p ≪ 1), M −1 has matrix element ∼ e np which leads to exponential increase of the error bar ∆g i with the qubit number n. To maintain the same error bar ∆g i , the number of repetitions N of the experiment eventually needs to increase exponentially with n. For practical applications, this exponential increase of N by the factor e np is typically not a problem for two reasons. First, as the detection error rate p is usually at a few percent level, the exponential factor e np remains moderate even for hundreds of qubits. Second, this exponential increase only applies when we need to measure each element of the distribution g i . In most of quantum information applications, we only need to measure certain operators which are expressed as tensor products of a constant number of Pauli operators for different qubits. In this case, N does not have the exponential increase as we show now.
Suppose we need to measure an operatorÔ, which is expressed asÔ = ⊗ n k=1 σ µ k k , where σ µ k k is a component of the Pauli matrices when µ k = 1, 2, 3 or the identity operator when µ k = 0. The number of the Pauli matrices n p in the tensor product expansion ofÔ is called the support ofÔ. To measure the operatorÔ, we choose the measurement basis to be the eigenbasis of σ µ k k for the kth qubit. In this measurement basis,Ô is diagonal with the matrix whereÔ i denotes the diagonal matrix element ofÔ. Therefore, by defining a corrected operatorÔ c , we can get the true expectation value Ô directly from the experimental data f j . Us- and for µ k = 0, diag(σ µ k k )D −1 k = [1,1]. For simplicity of notation, we take p 0 = p 1 = p. In this case, for µ k = 1, 2, 3, and the corrected operatorÔ c is related with the original operatorÔ by a simple scaling transformationÔ c = (1 − 2p) −npÔ . The scaling transformation is independent of the qubit number n, so the error bar of Ô does not have exponential increase with n when the operator O has a constant support n p . The scaling transformation also applies to collective operators, but some caution needs to be taken for calculation of their variance. For instance, if we take the collective spin operator J z ≡ n k=1 σ z k /2, it is easy to see that J c z = (1 − 2p) −1 J z as each of the terms of J z has support n p = 1. However, as J 2 z ≡ n/4 + k =l σ z k σ z l /4 which has non-uniform support for its superposition terms, one finds that J 2 . With this transformation, we can correct the distortion to the spin squeezing parameter by the detection error. Assume that the mean value of J is along the x-direction with J = J x and the squeezing is along the z-direction. The squeezing parameter is given by ξ = n J 2 z / J x 2 [4]. Using the transformation for J 2 z c and J c x , we find that where is the contribution to ξ 2 by the detection noise. After correction of the detection error, ξ c gets significantly smaller compared with ξ in particular when the qubit number n is large, and thus can be used to verify a much bigger entanglement depth using the criterion in Ref. [5]. From Eq. (10), we find that the variation ∆ξ c /∆ξ = ξ/ξ c . As typically ξ ≫ ξ c , the error bar for ξ c after correction of the detection error gets significantly larger, and we need to correspondingly increase the rounds of the experiment N to reduce the statistical error.
To illustrate application of the error correction method here, as an example, we apply it to detection of genuine multi-partite entanglement in graph states. For a graph state |G n of n qubits associated with a q-colorable graph G, the genuine n-party entanglement can be detected with the following witness operator [6] where Q l denotes the set of qubits with the lth color (l = 1, 2, · · · , q), I is the identity operator, and S k is the stabilizer operator for the kth qubit (which is a tensor product of the Pauli operators σ x k for the kth qubit and σ z k ′ for all it neighbors k ′ in the graph G). A state ρ has genuine n-qubit entanglement if tr (ρW Gn ) = W Gn < 0. For an ideal graph state, all its stabilizer operators S k have expect values S k = 1. With detection error, the value of S k gets significantly degraded. As an example, Fig. 1 shows the values of all S k for two particular 2-colorable graph states: a 10-qubit GHZ state (GHZ 10 ) and a linear cluster state (LC 10 ). We assume 3% detection error with p 0 = p 1 = p = 0.03 for each qubit. With a known magnitude p, the detection error can be easily corrected by a scaling transformation S c k = (1 − 2p) −n pk S k , where n pk is the support of the corresponding stabilizer operator S k . Fig. 1 shows that after error correction, S c k is almost unity. Its error bar increases a bit after the correction, but is still small. To show the influence on the entanglement detection, we assume the experimentally prepared graph state ρ ex corresponds to the ideal target state ρ id distorted by small depolarization noise independently acting on each qubit, so ρ ex = $ (ρ id ), where the noise super-operator $ = n k=1 $ k and $ k (ρ id ) = (1 − 3p n /4)ρ id + p n /4 µ=x,y,z σ (µ) k ρ id σ (µ) k [7]. In Fig. 2, we show the witness W Gn as a function of the preparation error rate p n , both before and after correction of the detection error (with an error rate p = 3%). For both GHZ 10 and LC 10 states, without correction of the detection error, we cannot detect any n-qubit entanglement even for a perfectly prepared state with p n = 0. After correction of the detection error, we can confirm genuine n-qubit entanglement as long as the preparation error p n 5%. So, correction of the detection error significantly improves the experimental performance, and the improvement gets more dramatic when the qubit number increases.
Finally, we briefly comment on the sensitivity of our error correction method to calibration of the detection error. In this method, the error magnitude p (or magnitudes p i , i = 0, 1, · · · , for general cases) is assumed to be known. If we have a relative error e in calibration of the magnitude p, .i.e., δp/p ∼ e, the scaling transformation on the detected operatorÔ leads to an relative error in the observed quantity δ Ô / Ô ∼ 2n p δp (1 − 2p) −1 ∼ 2n p pe. As long as 2n p p 1, which is typically the case as p ≪ 1, the relative error actually gets reduced and the method here can tolerate some uncertainty in calibration of the error magnitude p.
In summary, we have shown a method to correct any detection error through simple processing of the experimental data. The method applies to measurements in general many-particle settings, with or without separate addressing. Different from the conventional quantum error correction, this method does not require encoding or change of the quantum circuits and works under arbitrary magnitudes of the detection noise, as long as the error magnitude has been calibrated. The cost of this method is moderate as it only requires repetition of the same experiment by some additional rounds to gain enough statistics and thus the method can readily apply to many experimental settings.
Appendix:Proof of Eq (7). We can relate the L matrix to the M matrix defined in Eq. (3). Denote the space of n-bit binary strings with i bits of 1 as S i , and S i has dimension n i . The matrix element M σρ represents the probability of recording a n-bit binary string ρ as σ, and L ij is the probability of recording a signal ρ ∈ S j as any string in the S i space. As a collective measurement does not distinguish the binary strings in the same space S i , L ij is related to M σρ by L ij = σ∈Si M σρ .The probability L ij is apparently independent of the exact form of ρ, as long as ρ belongs to the space S j , so we can pick up any ρ ∈ S j in L ij = σ∈Si M σρ without alternation to the result of summation. From Eq. (4), we know M −1 µν = M µν (p ′ 0 , p ′ 1 ). Let us define where ν is an arbitrary element in S k . Now we show that N gives inverse of the matrix L: In the second line, we have changed the subscript ρ in M σρ to µ as both ρ, µ belong to S j . This proves Eq. (7) in the text. This work was supported by the NBRPC(973 Program) 2011CBA00300 (2011CBA00302), the IARPA MUSIQC program, the DARPA OLE program, the ARO and the AFOSR MURI program.