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Efficient Quantum State Tomography using the MERA in 1D critical system Presenter : Jong Yeon Lee (Undergraduate, Caltech)

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Quantum Tomography How to reconstruct quantum state? Black Box It prepares identical qubit state repeatedly Measure with Up/Down basis and +, – basis Reconstruct the state that black box prepares

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Quantum Tomography For generic case Black Box Measure with complete set of observables Reconstruct the state that black box prepares Obtain probability for observables Density Matrix

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Standard Method Problems of traditional method (Maximum likelihood estimation) – Number of Observables increases exponentially with the system size – Given that each observable has an uncertainty (as long as the number of measurement is not infinite), algorithms for finding optimal density matrix has also exponential time complexity. – The expectation value of observables decreases exponentially with the system size N – Any solution?

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Many quantum ground states have a special structure – “Area Law” : Entanglement Entropy related to Boundary – It takes a time exponentially large in system size for the quantum state to reach majority of Hilbert space by local interaction How to represent quantum states?

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Goal Minimize the number of measurements to experimentally identify 1D systems including critical system. Use variational family of states represented by type of tensor network states, which is called ‘MERA’ (having linear number of parameters with respect to the system size) ….. Density Matrix ….. Marcus et al. (2011)

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MERA Methodology developed by Guifre Vidal Originally exploited to find a quantum ground state of critical Hamiltonian Tree-like Structure – Isometry – Disentangler G.Evenbly, G.Vidal

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MERA tomography In MERA tomography, one finds a MERA circuit which well approximates a given experimental state Use ‘Partial’ Information to infer the state, therefore much more efficient that usual tomographic procedure. Following questions arise: 1.How to find parameters for MERA circuit? 2.What much more efficient? What is the Scaling of the number of measurements required? 3.Is reconstructed state actually close to given experimental state?

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1. MERA tomography – Principle Assume tensor network structure of state

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Choose disentanglers U such that sum of red box is minimized. Quantities in red box should be zero in ideal MERA circuit. Isometry W is automatically determined to diagonalized reduced density matrix 1. MERA tomography – Principle

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Performance Convergence of Objective Function in 24 Qubits This is the sum of probability weights on the wrong subspace, which should be zero in ideal case

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In standard method, the number of measurements required with fixed precision increases exponentially with system size. Theoretically, MERA state has only polynomial (square) increase of the number of parameters with system size. What would be the exact scaling for the number of measurements for various 1D critical systems? 2. Scaling of the number of measurements

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Renormalization of observables 8-sites observable is effected mapped into 4-sites observable

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To maintain same precision at all renormalized layers, the number of measurements should increase to compensate increased factors. 2. Scaling of the number of measurements Renormalize

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How to achieve smaller scaling factor? – Longest Residual Vector selection method – Out of all possible observables, choose a set of observable that most efficiently in tomographic procedure 4 6 operators 4 4 operators 2. Scaling of the number of measurements

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Scaling Factor for Measurements Precision of each observable measured 100 times

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Now we know that our MERA tomographic procedure works efficiently. However, whether or not a reconstructed state well approximates the given experimental state needs to be examined. Can it be verified during the tomographic procedure? 3. Verification : Fidelity between Actual versus Reconstructed states

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Upper bound for the fidelity between reconstructed state and given experimental state can be obtained using the quantity obtained during tomographic optimization procedure!

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Conclusion Superior optimization scheme Provided a bound for the distance between the experimental state and the reconstructed state using local data – Providing an efficient scalable certification method for critical states. Examined the performance of MERA tomography on the ground states of several 1D critical models.

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