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Quantum algorithms in the presence of decoherence: optical experiments Masoud Mohseni, Jeff Lundeen, Kevin Resch and Aephraim Steinberg Department of Physics, University of Toronto Friendly neighborhood theorists: Daniel Lidar, Sara Schneider,... Helpful summer student: Guillaume Foucaud

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Motivation Photons are an ideal system for carrying quantum info. (Nonscalable) linear-optics quantum computation may prove essential as part of quantum communications links. Efficient (scalable) linear-optical quantum computation is a very promising avenue of research, relying on the same toolbox (and more). In any quantum computation scheme, the smoky dragon is decoherence and errors. Without error correction, quantum computation would be nothing but a pipe dream. We demonstrate how decoherence-free subspaces (DFSs) may be incorporated into a prototype optical quantum algorithm.

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Prototype algorithm: Deutsch's Problem (2-qbit version) An oracle takes as input a bit x, and calculates an unknown one-bit function f(x). [quantum version: inputs x&y; outputs x & y f(x)] Our mission, should we decide to accept it: Determine, with as few queries as possible, whether or not f(0) = f(1). Classically: must measure both f(0) and f(1). [For n-bit extension, need at least 2 n-1 +1 queries] Quantum mechanically: a single query suffices. [Even for n-bit problem, since only yes/no outcome desired.]

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Standard Deutsch-Jozsa Algorithm H H xx H y y f(x) Physical realization of qubits We use a four-rail representation of our two physical qubits and encode the logical states 00, 01, 10 and 11 by a photon traveling down one of the four optical rails numbered 1, 2, 3 and 4, respectively. 1 2 3 4 Photon number basis 1 st qubit 2 nd qubit Computational basis Bob (oracle)Alice DJ algorithm and 4-rail qubits [Cf. Cerf, Adami, & Kwiat, PRA 57, R1477 (1998)]

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Quantum gateFour rails implementation 50/50 beam splitters swap between two rails Quantum gateFour rails implementation It is easy to implement a universal set up of one and two qubit operations in such a representation Implementation of simple gates

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Balanced oracle-01 f(0)=0,f(1)=1 00 11 10 01 00 11 10 01 00 11 10 01 Constant oracle-00 f(0)=f(1)=0 00 11 10 01 00 11 10 01 X Constant oracle-11 f(0)=f(1)=1 00 11 10 01 00 11 10 01 XX Balanced oracle-10 f(0)=1,f(1)=0 00 11 10 01 The transformations introduced by the 4 possible functions or “oracles” can also be implemented in this representation. Implementation of the oracle

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00 01 e i 01 11 10 e i 10 But after oracle, only qubit 1 is needed for calculation. Encode this logical qubit in either DFS: (00,11) or (01,10). Error model and decoherence-free subspaces Consider a source of dephasing which acts symmetrically on states 01 and 10 (rails 2 and 3)… Modified Deutsch-Jozsa Quantum Circuit y f(x) H xx H y H DFSs: see Lidar, Chuang, Whaley, PRL 81, 2594 (1998) et cetera. Implementations: see Kwiat et al., Science 290,498 (2000) and Kielpinski et al., Science 291, 1013 (2001).

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Schematic diagram of D-J interferometer 1234 1 2 3 4 12 3 4 Oracle 00 01 10 11 “Click” at either det. 1 or det. 2 (i.e., qubit 1 low) indicates a constant function; each looks at an interferometer comparing the two halves of the oracle. Interfering 1 with 4 and 2 with 3 is as effective as interfering 1 with 3 and 2 with 4 -- but insensitive to this decoherence model. Schematic of DJ, some failures

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3 4 1 2 1 2 4 23 Experimental Setup Oracle Swap Preparation Random Noise Mirror Waveplate Phase Shifter PBS Detector A B C D 3 4 3 4 DJ experimental setup

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CBCCCBBB DFS Encoding Original encoding Constant function Balanced function C B DJ without noise -- raw data

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CBCCCBBB DFS Encoding Original encoding C B Constant function Balanced function DJ without noise -- results

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CBCCCBBB DFS Encoding Original Encoding C B Constant function Balanced function DJ with noise-- results

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Coming Attractions: Non-orthogonal State Discrimination Non-orthogonal quantum states cannot be distinguished with certainty. This is one of the central features of quantum information which leads to secure (eavesdrop-proof) communications. Crucial element: we must learn how to distinguish quantum states as well as possible -- and we must know how well a potential eavesdropper could do. (work with J. Bergou et al.)

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Theory: how to distinguish non- orthogonal states optimally Step 1: Repeat the letters "POVM" over and over. The view from the laboratory: A measurement of a two-state system can only yield two possible results. If the measurement isn't guaranteed to succeed, there are three possible results: (1), (2), and ("I don't know"). Therefore, to discriminate between two non-orth. states, we need to use an expanded (3D or more) system. To distinguish 3 states, we need 4D or more. Step 2: Ask Janos, Mark, and Yuqing for help.

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But a unitary transformation in a 4D space produces: …and these states can be distinguished with certainty up to 55% of the time A test case Consider these three non-orthogonal states: Projective measurements can distinguish these states with certainty no more than 1/3 of the time. (No more than one member of an orthonormal basis is orthogonal to two of the above states, so only one pair may be ruled out.)

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Experimental layout (ancilla)

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Success! The correct state was identified 55% of the time-- Much better than the 33% maximum for standard measurements. "I don't know" "Definitely 3" "Definitely 2" "Definitely 1"

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Summary We have demonstrated the utility of decoherence-free subspaces in a prototype linear-optical quantum algorithm. The introduction of localized turbulent airflow produced a type of “collective” optical dephasing, leading to large error rates. With the DFS encoding, the error rate in the presence of noise was reduced to 7%, essentially its pre-noise value. We note that the choice of a DFS may be easier to motivate via consideration of the physical system than from purely theoretical (quantum circuit) considerations! More recent results: successfully distinguish among 3 non- orthogonal states 55% of the time, where standard quantum measurements are limited to 33%. Also: "state filtering" or discrimination of mixed states.

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