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True experimental reconstruction of quantum states and processes via convex optimization

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Abstract

We use a constrained convex optimization (CCO) method to experimentally characterize arbitrary quantum states and unknown quantum processes on a two-qubit NMR quantum information processor. Standard protocols for quantum state and quantum process tomography are based on linear inversion, which often result in an unphysical density matrix and hence an invalid process matrix. The CCO method, on the other hand, produces physically valid density matrices and process matrices, with significantly improved fidelity as compared to the standard methods. We use the CCO method to estimate the Kraus operators and characterize gates in the presence of errors due to decoherence. We then assume Markovian system dynamics and use a Lindblad master equation in conjunction with the CCO method, to completely characterize the noise processes present in the NMR system.

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Acknowledgements

All the experiments were performed on a Bruker Avance-III 600 MHz FT-NMR spectrometer at the NMR Research Facility of IISER Mohali.

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  1. Department of Physical Sciences, Indian Institute of Science Education and Research (IISER) Mohali, Sector 81 SAS Nagar, Sahibzada Ajit Singh Nagar, Punjab, 140306, India

    Akshay Gaikwad,  Arvind & Kavita Dorai

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  1. Akshay Gaikwad
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A Kraus operators

A Kraus operators

The complete set of valid Kraus operators for the two-qubit system have been experimentally computed using the CCO QPT method. The Kraus operators corresponding to the Identity, CNOT gate and control-\(R^{\pi }_{x}\) gate are given below:

  • Kraus operators corresponding to Identity gate

    $$\begin{aligned} E_1= & {} {\begin{bmatrix} -0.0308 + 0.0457i &{} \quad -0.0028 - 0.0077i &{} \quad 0.0626 + 0.1056i &{} \quad 0.0022 + 0.0078i \\ 0.0070 + 0.0095i &{} \quad -0.0393 + 0.0633i &{} \quad -0.0055 - 0.0060i &{} \quad 0.0550 + 0.1068i \\ 0.0755 - 0.0678i &{} \quad 0.0203 + 0.0042i &{} \quad 0.0279 - 0.0575i &{} \quad 0.0052 + 0.0001i \\ 0.0395 + 0.0187i &{} \quad 0.0850 + 0.0079i &{} \quad 0.0153 + 0.0005i &{} \quad 0.0451 - 0.0399i \end{bmatrix}} \\ E_2= & {} { \begin{bmatrix} 0.0571 + 0.0943i &{} \quad -0.0133 + 0.0201i &{} \quad -0.1932 - 0.0604i &{} \quad -0.0069 - 0.0085i \\ -0.0071 + 0.0271i &{} \quad -0.0082 + 0.0968i &{} \quad 0.0173 - 0.0019i &{} \quad -0.1724 - 0.0574i \\ 0.0189 - 0.1154i &{} \quad 0.0269 - 0.0103i &{} \quad -0.0281 - 0.1005i &{} \quad 0.0091 - 0.0038i \\ 0.0481 - 0.0087i &{} \quad 0.0723 - 0.0485i &{} \quad -0.0040 + 0.0067i &{} \quad -0.0220 - 0.0980i \end{bmatrix}} \\ E_3= & {} { \begin{bmatrix} -0.0442 - 0.9758i &{} \quad -0.0014 + 0.0418i &{} \quad 0.0103 + 0.0259i &{} \quad -0.0100 - 0.0007i \\ 0.0005 - 0.0271i &{} \quad -0.0550 - 0.9813i &{} \quad 0.0095 + 0.0029i &{} \quad 0.0129 + 0.0272i \\ 0.0101 - 0.0239i &{} \quad 0.0088 - 0.0002i &{} \quad -0.0190 - 0.9617i &{} \quad 0.0233 + 0.0404i \\ -0.0096 + 0.0017i &{} \quad 0.0101 - 0.0205i &{} \quad 0.0242 - 0.0412i &{} \quad 0.0021 - 0.9671i \\ \end{bmatrix}} \end{aligned}$$

  • Kraus operators corresponding to CNOT gate

    $$\begin{aligned} E_1= & {} {\begin{bmatrix} 0.0344 - 0.0042i &{} \quad 0.0389 + 0.0130i &{} \quad -0.0068 + 0.0035i &{} \quad -0.0691 - 0.0003i \\ -0.0039 - 0.0038i &{} \quad -0.0208 - 0.0054i &{} \quad -0.0494 + 0.0543i &{} \quad 0.0125 + 0.0320i \\ 0.0548 + 0.0194i &{} \quad -0.0023 - 0.0251i &{} \quad -0.0714 + 0.0137i &{} \quad -0.0094 - 0.0117i \\ 0.0208 + 0.0094i &{} \quad 0.0654 + 0.0264i &{} \quad -0.0162 + 0.0148i &{} \quad 0.0221 + 0.0124i \end{bmatrix}} \\ E_2= & {} {\begin{bmatrix} 0.0124 + 0.0245i &{} \quad 0.0065 - 0.0008i &{} \quad -0.0508 - 0.1283i &{} \quad -0.0079 + 0.0205i \\ 0.0727 + 0.0709i &{} \quad -0.0132 - 0.0155i &{} \quad -0.0941 + 0.0168i &{} \quad 0.0494 - 0.0326i \\ 0.0323 - 0.0020i &{} \quad -0.0552 + 0.0537i &{} \quad 0.1017 + 0.0139i &{} \quad -0.0577 - 0.0248i \\ 0.0400 + 0.0811i &{} \quad -0.0112 + 0.0281i &{} \quad 0.0603 + 0.0204i &{} \quad -0.0381 - 0.0261i \end{bmatrix}} \\ E_3= & {} {\begin{bmatrix} 0.0907 - 0.0140i &{} \quad -0.0599 + 0.0491i &{} \quad 0.0581 + 0.0467i &{} \quad 0.0292 + 0.0058i \\ -0.0567 + 0.0142i &{} \quad -0.0978 - 0.0171i &{} \quad 0.0109 + 0.0093i &{} \quad -0.0310 - 0.1036i \\ 0.0267 + 0.0135i &{} \quad -0.0221 + 0.0546i &{} \quad -0.0700 + 0.0595i &{} \quad -0.0752 - 0.0404i \\ -0.0269 - 0.0463i &{} \quad -0.0340 + 0.0427i &{} \quad 0.0765 + 0.0205i &{} \quad -0.1294 + 0.0564i \end{bmatrix}} \\ E_4= & {} { \begin{bmatrix} 0.1786 + 0.0344i &{} \quad 0.1327 - 0.0629i &{} \quad 0.0228 + 0.0866i &{} \quad -0.0018 - 0.0201i \\ 0.0052 - 0.0290i &{} \quad -0.1264 - 0.0353i &{} \quad 0.0174 - 0.0797i &{} \quad -0.0397 + 0.0932i \\ -0.0346 + 0.0199i &{} \quad 0.0383 - 0.0361i &{} \quad 0.1008 - 0.0337i &{} \quad -0.1466 - 0.0222i \\ -0.0024 - 0.0169i &{} \quad -0.0415 - 0.0293i &{} \quad 0.1058 - 0.0050i &{} \quad 0.1214 - 0.1034i \end{bmatrix}} \\ E_5= & {} {\begin{bmatrix} 0.0706 + 0.9517i &{} \quad -0.0369 + 0.0847i &{} \quad 0.0250 + 0.0166i &{} \quad -0.0245 - 0.0130i \\ -0.0139 - 0.1052i &{} \quad -0.1412 + 0.9412i &{} \quad -0.0442 - 0.0280i &{} \quad -0.0077 - 0.0040i \\ -0.0187 + 0.0169i &{} \quad -0.0218 - 0.0073i &{} \quad -0.0414 + 0.0215i &{} \quad -0.0410 + 0.9380i \\ -0.0065 - 0.0224i &{} \quad -0.0537 + 0.0269i &{} \quad 0.0297 + 0.9390i &{} \quad -0.0516 + 0.0110i \end{bmatrix}} \end{aligned}$$

  • Kraus operators corresponding to control-\(R^{\pi }_{x}\) gate

    $$\begin{aligned} E_1= & {} { \begin{bmatrix} 0.0012 + 0.0210i &{} \quad 0.0165 + 0.0134i &{} \quad -0.0744 + 0.0001i &{} \quad -0.0121 - 0.0364i \\ 0.0359 + 0.0089i &{} \quad 0.0251 - 0.0080i &{} \quad -0.0654 - 0.0831i &{} \quad -0.0097 - 0.0251i \\ -0.0234 + 0.0153i &{} \quad -0.0354 - 0.0211i &{} \quad -0.0461 + 0.0131i &{} \quad 0.0004 + 0.0068i \\ 0.0365 + 0.0369i &{} \quad -0.0233 + 0.0167i &{} \quad -0.0068 + 0.0143i &{} \quad 0.0125 - 0.0159i \end{bmatrix}} \\ E_2= & {} { \begin{bmatrix} 0.0153 + 0.0286i &{} \quad 0.0001 - 0.1142i &{} \quad -0.0595 - 0.0245i &{} \quad 0.0153 - 0.0085i \\ -0.0141 + 0.0290i &{} \quad -0.0039 - 0.0323i &{} \quad -0.0097 + 0.0340i &{} \quad 0.1150 + 0.0166i \\ 0.0058 + 0.0004i &{} \quad -0.0092 - 0.0963i &{} \quad 0.0556 + 0.0204i &{} \quad -0.0167 - 0.0196i \\ -0.0192 + 0.0136i &{} \quad -0.0308 + 0.0298i &{} \quad 0.0407 + 0.0222i &{} \quad -0.1054 - 0.0405i \end{bmatrix}} \\ E_3= & {} { \begin{bmatrix} 0.1537 + 0.0345i &{} \quad 0.0717 + 0.0257i &{} \quad 0.0169 - 0.0805i &{} \quad -0.0006 - 0.0094i \\ -0.0074 + 0.0235i &{} \quad -0.1425 - 0.0207i &{} \quad -0.0232 + 0.0145i &{} \quad 0.0001 - 0.0276i \\ 0.0178 + 0.0428i &{} \quad -0.0375 - 0.0168i &{} \quad 0.0243 - 0.0328i &{} \quad -0.0154 + 0.1688i \\ -0.0239 - 0.0132i &{} \quad -0.0232 + 0.0146i &{} \quad 0.0017 - 0.1398i &{} \quad 0.0512 - 0.0880i \end{bmatrix}} \\ E_4= & {} { \begin{bmatrix} 0.0686 - 0.0160i &{} \quad -0.1101 - 0.0036i &{} \quad -0.0419 - 0.0764i &{} \quad -0.0257 - 0.0232i \\ -0.1221 - 0.0570i &{} \quad -0.0450 + 0.0133i &{} \quad -0.0657 + 0.0230i &{} \quad 0.0491 + 0.0496i \\ 0.0651 + 0.0241i &{} \quad 0.0643 - 0.0430i &{} \quad -0.1377 + 0.1614i &{} \quad -0.0061 + 0.0345i \\ -0.0239 + 0.0021i &{} \quad -0.0537 - 0.0392i &{} \quad -0.0594 - 0.0210i &{} \quad 0.0213 + 0.1907i \end{bmatrix}} \\ E_5= & {} {\begin{bmatrix} 0.1841 + 0.9399i &{} \quad -0.0704 + 0.1026i &{} \quad 0.0143 + 0.0058i &{} \quad 0.0012 + 0.0004i \\ -0.0949 - 0.0906i &{} \quad 0.0979 + 0.9445i &{} \quad 0.0084 + 0.0134i &{} \quad -0.0049 + 0.0210i \\ 0.0077 - 0.0108i &{} \quad -0.0075 + 0.0042i &{} \quad -0.0249 + 0.0765i &{} \quad 0.9336 - 0.0790i \\ -0.0076 - 0.0216i &{} \quad -0.0092 - 0.0086i &{} \quad 0.9304 - 0.0835i &{} \quad 0.0338 + 0.0811i \end{bmatrix}} \end{aligned}$$

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Gaikwad, A., Arvind & Dorai, K. True experimental reconstruction of quantum states and processes via convex optimization. Quantum Inf Process 20, 19 (2021). https://doi.org/10.1007/s11128-020-02930-z

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