Quantum Computation and Information

Quantum computation and information broadly encompasses the study of how quantum systems can be used for information processing and also how ideas from information theory can be adapted to understand quantum systems and their behavior.

In quantum information processing protocols, information can be encoded into the states of a suitable quantum system and processing is achieved by evolving the system in a controlled fashion according to the laws of quantum mechanics. Finally a measurement on the system yields outcomes which are then translated into a classical outcome for the protocol. Clever use of characteristic features of quantum mechanics such as superposition and entanglement provides a form of parallel information processing, which can in theory give advantages over any competing classical protocol.

Examples of quantum information processing protocols include quantum computation and quantum cryptography. Quantum computation offers algorithms that in principle solve particular classical problems, such as prime factorization, much faster than is possible with any classical computing device. Quantum cryptography offers protocols for sharing cryptographic keys securely and this has been explored in various practical devices.

Separately, ideas from classical information theory have been adapted to describe the information content, in terms of processing powers, of quantum states and systems.

Quantum Parameter Estimation and Unitary and State Discrimination

The inherent statistical nature of quantum physics has profound implications for issues such as parameter estimation and state and unitary discrimination. Parameter estimation entails inferring a parameter, such as a phase shift or magnetic field strength, for a process whose nature, but not strength is known. The statistical nature of measurement outcomes in quantum theory implies that one can seldom establish the values of such parameters without some uncertainty. It is possible to establish general bounds on this uncertainty under any conceivable measurement and estimation scenario. The quantum Fisher information is the quantity which governs these bounds and this can be calculated exactly in some circumstances. Remarkably, these calculations show that one can often use special features quantum theory to attain precision in estimation which is unavailable to classical systems.

My current research focuses on assessing various estimation schemes for phase shift, bit-flip and depolarizing operations particularly when the available system states are noisy. I have shown that when the system is initially in a highly mixed state, such as those encountered in solution-state NMR, correlations between the system can assist in improving estimation accuracy although these advantages disappear when the initial states are pure (see "Depolarizing channel parameter estimation using noisy initial states, Preprint arXiv:1506.06147 (2014) and "Mixed-state Pauli channel parameter estimation," Phys. Rev. A 87, 032301 (2013)).

Previously my collaborators and I found the optimal estimation scenario for estimating the depolarizing channel parameter for arbitrary dimensions and showed that it is attained with a maximally entangled state (see "Probing the qudit depolarizing channel," J.Phys. A: Math. Theor. 44, 205306 (2011)).

Quantum state and unitary discrimination considers the extent to which it is possible to distinguish between a collection of specified quantum states or evolution operations. I showed that it is possible to regard the Deutsch-Jozsa algorithm as a device for unitary transformation discrimination and applied this to bound the performance of the algorithm when using mixed input states. Details are available at "Discrimination of unitary transformations in the Deutsch-Jozsa algorithm: Implications for thermal-equilibrium-ensemble implementations," Phys. Rev. A 81 052323 (2010).

Expectation Value Quantum Computing

Quantum algorithms are usually formulated for execution on a quantum computer consisting of a single quantum system. The computation begins by initializing the system in a pure state and ends with a projective measurement, whose outcome yields a meaningful problem solution with good probability. In contrast the most successful experimental realizations of quantum algorithms to date have used room-temperature, solution state nuclear magnetic resonance (NMR). The NMR sample contains an ensemble of identical, non-interacting quantum computers. Such systems are in highly mixed states and measurements on these can only yield expectation values of observables, and not the standard projective measurement outcomes. Quantum algorithms have been modified accordingly. However, these modifications (e.g. pseudopure state preparation) have been designed to mimic the original algorithm as much as possible and have not used the ensemble to any advantage.

I demonstrated that it is possible to use the ensemble to speed up Grover's search algorithm on an expectation value quantum computers (see Phys. Rev. A. 65, 052321 (2002)). The speedup is by a factor independent of the database size and depends on measurement resolution.

NMR Realization of the Deutsch-Jozsa algorithm

NMR has become a favorite tool for demonstrating the dynamics of quantum computation. While working at North Carolina State University, I lead an experimental NMR implementation of the Deutsch-Jozsa algorithm for functions with three bit arguments. We showed that it is at this level that characteristic features of quantum mechanics first become essential in the algorithm. We used the three carbon spins of  13C labeled alanine (CH3CH(HN2)CO2H) for qubits and successfully applied the algorithm to a representative of each class of balanced functions at this level (see Phys. Rev. A 62, 022304 (2000)). More details can be found in the preprint quant-ph/9910006. Output spectra for representatives of each function class are available via the following links:

  f1(x2,x1,x0) = x2  f1 spectrum (PostScript)
  f2(x2,x1,x0) = x2 + x1  f2 spectrum (PostScript)
  f3(x2,x1,x0) = x2 + x1 + x0  f3 spectrum (Postscript)
  f4(x2,x1,x0) = x2x1 + x0  f4 spectrum (PostScript)
  f5(x2,x1,x0) = x2 x1 + x2 + x0  f5 spectrum (PostScript)
  f6(x2,x1,x0) = x2 x1 + x2 + x1 + x0  f6 spectrum (PostScript)
  f7(x2,x1,x0) = x2 x1 + x1 x0 + x2 + x1  f7 spectrum (PostScript)
  f8(x2,x1,x0) = x2 x1 + x1 x0 + x2  f8 spectrum (PostScript)
  f9(x2,x1,x0) = x2 x1 + x1 x0 + x2 x0  f9 spectrum (PostScript)
  f10(x2,x1,x0)= x2 x1 + x1 x0 + x2 x0 + x1 + x0  f10 spectrum (PostScript)

Our implementation incorporated an "indirect" realization of a two qubit gate between two spins, using a chain of couplings between the spins rather than the "direct" realization which uses only the coupling between the spins. Such indirect gate realizations are expected to become indispensable for quantum computation as the number of qubits increases.

Quantum Computation and Information Publications

  1. David Collins and Justin Endicott, "Connecting optical intensities and electric fields using a triple interferometer," arXiv:2008.12641 (2020).
  2. David Collins, "Qubit-channel metrology with very noisy initial states," Phys. Rev. A 99, 012123 (2019). Preprint arXiv:1706.03552 (2017).
  3. David Collins and Jaimie Stephens, "Depolarizing-channel parameter estimation using noisy initial states," Phys. Rev. A 92, 032324 (2015). Preprint arXiv:1506.06147 (2015).
  4. David Collins, "Mixed-state Pauli channel parameter estimation," Phys. Rev. A 87, 032301 (2013). Preprint arXiv:1208.6049 (2012).
  5. Michael Frey, David Collins, and Karl Gerlach, "Probing the qudit depolarizing channel," J. Phys. A: Math. Theor. 44, 205306 (2011). Preprint arXiv:1008.1557 (2010).
  6. David Collins, "Discrimination of unitary transformations in the Deutsch-Jozsa algorithm: Implications for thermal-equilibrium-ensemble implementations," Phys. Rev. A 81 052323 (2010). Preprint arXiv:1002.4227 (2010).
  7. Michael Frey and David Collins, "Quantum Fisher information and the qudit depolarization channel," Quantum Information and Computation VII, Eds. Donker, Pirich and Brandt, Proc. SPIE, 7342, 73420N (2009).
  8. David Collins, "Discrimination of Unitary Transformations and Quantum Algorithms," Quantum Communication, Measurement and Computing (QCMC): Ninth International Conference on QCMC, AIP Conf. Proc., 1110, 403 (2009). Preprint arXiv:0811.1359. (2008).
  9. Tomasz M. Kott and David Collins, "Statistical comparison of ensemble implementations of Grover's search algorithm to classical sequential searches," Phys. Rev. A 77, 052314 (2008). Preprint quant-ph/0708.0808 (2007).
  10. Brandon M. Anderson and David Collins, "Polarization Requirements for Ensemble Implementations of Quantum Algorithms with a Single Bit Output," Phys. Rev. A 72, 042337 (2005). Preprint quant-ph/0508061.
  11. Arvind and David Collins, "Scaling issues in ensemble implementations of the Deutsch-Jozsa algorithm," Phys. Rev. A 68, 052301 (2003). Preprint quant-ph/0307153.
  12. David Collins, "Shortening Grover's search algorithm for an expectation value quantum computer," Preprint quant-ph/0209148. Published in the proceedings of the QCMC'02 conference (2002).
  13. David Collins, "Modified Grover's algorithm for an expectation-value quantum computer," Phys. Rev. A. 65, 052321 (2002). Preprint quant-ph/0111108.
  14. David Collins, K. W. Kim, W. C. Holton, H. Sierzputowska-Gracz, E. O. Stejskal, "Orchestrating an NMR quantum computation: the N=3 Deutsch-Jozsa algorithm," Preprint quant-ph/0105045 (2001).
  15. David Collins, K. W. Kim, W. C. Holton, H Sierzputowska-Grazc, and E. O. Stejskal, "NMR quantum computation with indirectly coupled gates," Phys. Rev. A 62, 022304 (2000).
  16. David Collins, K. W. Kim, and W. C. Holton, "Deutsch-Jozsa algorithm as a test of quantum computation," Phys Rev A, 58, 1633 (1998).

Selected Conference Presentations and Posters

  1. "Enhanced Noisy Depolarizing Channel Parameter Estimation," Poster presented at the 9th Conference on the Theory of Quantum Computation, Communication and Cryptography (TQC 2014), Singapore, May 2014.
  2. "Correlated Quantum States and Enhanced Mixed State Pauli Channel Parameter Estimation," Poster presented at the International Conference on Quantum Information and Quantum Computing, Bangalore, India, January 2013.
  3. "No Advantage to Entanglement in Bit-Flip Parameter Estimation," Poster presented at the International Conference on Quantum Information, Ottawa, Canada, June 2011.
  4. "Optimal Estimation of Single Qubit Quantum Evolution Parameters," APS March Meeting, Portland, Oregon, March 2010.
  5. "Discrimination of Unitary Transformations and Quantum Algorithms," 9th International Conference on Quantum Communication, Measurement and Computing, Calgary, Canada, August 2008.
  6. "Scaling Issues in Ensemble Quantum Algorithms," Poster presented at the Quantum Information and Quantum Control Conference, Fields Institute, Toronto, July 2004.
  7. "Could Quantum Computing Aid Path Integration?," MSRI conference, Berkeley, California, December 2002.
  8. "Shortening Grover's Search Algorithm for an Expectation Value Quantum Computer," Poster presented at the 6th International Conference on Quantum Communication, Measurement and Computing, MIT, Cambridge, Massachusetts, July 2002.  Postscript
  9. "NMR Quantum Computation with Indirectly Coupled Gates," Poster presented at the 41st Experimental Nuclear Magnetic Conference, Pacific Grove, California, 2000.  Postscript

Invited Talks

  1. "Quantum Computing with Ensembles: Strange Physics for Ordinary Tasks," April 2006.   Pdf
  2. "Quantum Computing with Ensembles," Talk at Bucknell University, November 2005.   Pdf

Functional Integration

Publications

  1. David Collins, "Two-State Quantum Systems Interacting with Their Environments: A Functional Integral Approach," PhD thesis, University of Texas at Austin, (December 1997).  (Postscript available on request)

This page is maintained by David Collins
Last modified: 3 February 2019