Report

Optical Communications at the Quantum Limit Zachary Dutton Raytheon BBN Technologies Cambridge, MA ICQIQC 2013 IISc Bangolore, India 11 January, 2013 Quantum Information Processing (QuIP group) Founded at BBN in 2008 Now comprised of researchers in superconducting electronics, experimental quantum optics, information and physics theory Identify new discoveries in physics laboratories that can deliver new capabilities to computation, communication, and sensing systems Understand the impacts of these discoveries on enhanced (or new) capabilities and transition them to engineered systems Interests range from physical to application layer Partner with academic and industrial partners to attack the difficult challenges Applications (e.g. Entanglement enhanced LADAR; secret key distribution, quantum computation) System tools Devices (e.g. quantum repeater stations) Components (e.g. multi-qubit chips, entanglement sources) (e.g. quantum functional language and compilation ) Physical Theory (e.g. non-standard optical receivers) Physical Technologies (e.g. superconducting qubits, superconducting photon detectors, SPDC sources) Application areas • Quantum computation – cQED based superconducting qubits and processing architectures – Programming languages, error correction, and compilers for quantum computation – Low power high performance classical computation based on superconducting logic • Quantum communications – Long distance quantum key distribution (QKD) – Photon Information Efficient COMMunications (PIECOMM) fundamental limits of power & spectral efficiency in communication • Quantum enhanced sensing – Quantum imaging fundamental limits of imaging power efficiency – Quantum illumination using entanglement for improved target detection BBN Laboratory for Bits and Waves cryo-lab • Research in superconducting circuit based quantum computation – In collaboration with IBM and university partners we are designing a scalable cQED qubit architecture based on the surface code Surface code structure • Two 10 mk dilution refrigerators Single-qubit Cliffords characterized with RB Q1 Q2 Q3 Gate time 20 ns 16.7 ns 16.7 ns Avg. error 0.0035 0.0025 0.0030 Error per generator 0.0022 0.0016 0.0019 3-qubit / 2-cavity sub-cell E. Magesan, et. al PRL 109, 080505 (2012) • Research on nonlinear optical effects at microwave frequencies – Coherent population trapping (CPT) K. Murali, et. al PRL 93, 087003 (2004) W. R. Kelly, et. al, PRL 104 163601 (2010) Light is very good at carrying information – Much higher bandwidth (higher data rates) than RF – Long range (suitable for deep space communication) Holevo limit Homodyne Heterodyne – What are the limits of optical communication and how can we reach them? – It requires quantum theory to model a physical channel Our “PIECOMM” team BBN MIT • Saikat Guha (PI) • Jeffrey H. Shapiro – Nivedita Chandrasekaran • Zachary Dutton • Hari Krovi • Monika Patel • Jeff Chen • Seth Lloyd • Lizhong Zheng – Hye Won Chung • Gregory Wornell – Yuval Kochman – Ligong Wang • Franco N. C. Wong – Valentina Schettini • Jonathan Habif • Richard Lazarus • Karl Berggren – Francesco Bellei – Hasan Korre Discrimination of optical states below the standard quantum limit (SQL) Direct detection of optical states • Direct detection (intensity detection): the “shot noise” limit – Photon detection (arrival) process is a Poisson point process with rate, – Total number of “clicks” on the detector (k) has a Poisson distribution – The number of clicks detected has no information about phase ϕ • The “phase space” picture Im Re Coherent detection of optical states • Coherent detection (Homodyne and Heterodyne detection) measures both intensity and phase – Homodyne detection measures one chosen quadrature (θ) – Heterodyne detection measures two orthogonal quadratures simultaneously but with twice as much noise on each measurement • Quantum description of a classical laser pulse: coherent state of light Detection scheme matched to modulation • On-off keying (OOK) modulation Im Direct detection Re • Binary phase shift keying (BPSK) modulation Im Homodyne detection Re • Quadrature phase shift keying (QPSK) modulation Im Heterodyne detection Re Quantum State Discrimination • Direct, homodyne, and heterodyne detection define the standard quantum limit (SQL) • However, there exists a lower fundamental limit (Helstrom bound) to discrimination error for non-orthogonal states • Optimal (Minimum Probability of Error) discrimination between two non-orthogonal pure states – Binary pure states, and – MPE measurement: Minimize such that – Helstrom measurement , with Prob. and – Minimum error probability Helstrom(1976) The Dolinar receiver • Can achieve the binary Helstrom bound • Utilizes real time classical feedback and nulling • Can beat the SQL limit (coherent and DD) • Original demonstration beat DD limit (with QE corrected) on binary phase shift keyed (BPSK) input Cook, Martin, Geremia Nature (2007) Dolinar (1976) • Recently NIST-Gaithersburg demonstrated a QPSK version (Boundarant receiver) with an amplitude slicing technique (Takeoka receiver) – 13 dB below SQL (6 dB below perfect detector SQL) Becerra, et. al, Nature Phonics (in press) The “Generalized Kennedy” Receiver click “off” say “off” say “on” “on” no click • Utilizes optimized (but constant) nulling • Recently demonstrated by NICT and unambiguously beats SQL Exact nulling Tsujino, PRL (2011) arXiv:1103.5592 Pe (OOK) DD GK Helstrom Np(dB) PPM demodulation using the Conditional Pulse Nulling (CPN) receiver PPM Pulses SPD Decision Nulling 1 1 k k DD pulse 2, 3, 2 1 1 4 2 k Null pulse-1 k 2 DD pulse 3, 4 and DD 2 3 Null pulse-2 4 3 DD pulse 4 and DD Null pulse-3 3 4 and DD 3 4 Nulling “Decision Tree” Reaches sub-SQL error demodulation of a codeword of multiple symbols (joint detection receiver – JDR) Dolinar, MIT Ph.D. Thesis 1976, TDA Progress Report, 42-72, 1982 Guha, Habif, Takeoka, J. Mod. Optics, Vol. 58, Nos. 3–4, 10–20, 257–265, 2011 Decoder System Diagram of a CPN Receiver Chen, Habif, Dutton, Lazarus & Guha, Nature Photonics (2012), Xiv:1111.4017 Experimental results • First experimentally tested perfect nulling • Beat direct detection limit by 1.3 dB at Np =0.65 • At higher Np the CPN receiver begins to degrade, in agreement with a model with a mode-mismatch of 0.05 • We believe this mis-match could be improved by an order of magnitute, which would yield 6 dB improvement at Np =2 DD CPN Helstrom Dashed: ideal Solid: model b a a q b Np (1 d ) N p e N null e iq iq N 0 | a b | N P D 2 [ 2 (1 cos q ) D T D P ] N P [( 1 d 1)( 1 cos q ) ( 2 d 2 1 d )] N P DTT T Optimal nulling CPN • Just as the Generalized Kennedy receiver (with optimal nulling amplitude) can improve upon direct detection, the CPN can be improved by optimal nulling – This effect is pronounced at low Np<1.0 – For Np=0.65, this ideally increases the improvement over DD from 1.3 dB to 2.2 dB. – For this data set, 3% mode mis-match model gave best agreement with data We demonstrated 2.1 dB improvement in Pe over DD by optimizing the null amplitude Chen, Habif, Dutton, Lazarus & Guha, Nature Photonics (2012), Xiv:1111.4017 Achieving the Helstrom limit generally • The CPN receiver approaches the Helstrom limit in a special case – PPM + high photon number – Recent work has extended to more generally to high photon number case Nair & Guha, arXiv:1212.2048 – Is implemented simply with linear optics and classical feedback • Is there a prescription to do this for all regimes? – Yes – though a clear implementation is not yet apparent – da Silva’s approach solves the long standing problem of minimum probability of error (MPE) measurement in discriminating an arbitrary number of coherent states da Silva, Guha and Dutton, QCMC 2012, arXiv: 1208.5758 (2012) The slicing receiver Infinite dimensional coherent states can be “compressed” into finite dimensional qubits U is a highly nonlinear operation Coherent states can be “sliced” into multiple small amplitude pieces (living in the |0> and |1> sub-space) then compressed into a qubit register The {Ui} are determined by unitary compression: Ul |hj>|mj,l> = |f>|mj,l> |{hj>} are hypotheses Binary example: BPSK receiver The compressed qubit da Silva, Guha and Dutton, QCMC’12, ArXiv: 1208.5758 (2012) Slicing receiver performance • The final step is then a projective measurement of the register • Note that this is essentially a small continuous variable optical quantum computer The receiver is seen here to work for 2 and 3 state discrimination and is fully generalizable 0.6 0.5 0.1 0.05 0.1 Pe Pe 0.01 0.005 0.001 0.0005 0.05 0.0001 0.00005 0.00001 0.01 n = 2, 10, 30, 100 slices 0.1 0.2 0.3 0.4 0.5 Α 1 n = 2, 10, 30, 100 slices 0.005 2 3 0.1 0.2 0.3 0.4 0.5 1 Α 2 3 4 5 Fundamental limits of capacity Connecting of sub-SQL to capacity • We have recently identified fundamental limits of photon efficiency (bits per photon) for classical optical communications – Quantum mechanics (Holevo bounds) must be employed to calculate – Important for power contrained systems such as deep space (e.g. lunar and Mars) systems • Classical coherent states (laser pulses) are sufficient to reach this capacity – This is good news, since any other state would be destroyed during transmission through the atmosphere – One needs non-standard receivers – Joint detecton receivers (JDRs) are necessary Giovannetti, Guha, Lloyd, Maccone, & Shapiro (2004) Shapiro, Guha, Erkmen (2005) Guha, PRL (2011) Reliable communication over a noisy channel Claude Shannon “Father of information theory” 1916-2001 Classical communication over a classical channel Classical information transmission via classical symbols noisy channel Noise does not preclude error-free digital communication Error-free communication can be accomplished if the data rate is below the channel capacity, appropriate error-correction coding is employed Channel capacity is the maximum mutual information Shannon (1948) Shannon’s intuition • Coding is essentially a way to drive down errors via redundancy – – e.g. a rate r=0.33, the repetition code would send “000” and “111” to communicate one bit during three pulse slots using a binary modulation alphabet Chooses 2k of the 2n possible codewords to form a rate r=k/n binary code [100100101] (decoding) [101101001] Codebook: a pruned set of 2nR binary sequences (Code rate, R = k/n) - Sending nR bits of information over n channel uses: R bits per channel use - Shannon [1948]: As long as R < C, there exists a (n,k,d) codebook with n large enough that the probability of codeword decoding error goes to zero as n goes to infinity. Concatenated coding and JDRs • Traditional low-complexity codes rely on concatenation of inner (binary) codes (e.g. BCH) and outer (non-binary) codes (e.g. Reed-Solomon). • Our JDR results (Cn > C1) are all Shannon capacity results, suggesting a concatenated coding approach where the JDR acts on the inner code inner super-channel: Shannon capacity: Cn > C1 (N,R) outer encoder (n,r) inner encoder Modulator Physical channel Optical Receiver Demodu -lator Detec -tor Inner decoder Outer decoder Joint-detection receiver (JDR) Guha, Dutton, Shapiro arXiv:1102.1963v1 [quant-ph] Proc. ISIT 2011 • In the CPN case we can use Reed-Solomon codes as outer codes over the PPM inner code • • • • Code parameters are the block length n and information bits per block k PPM rate rPPM=(log2M)/M The total rate of the concatenated code is r=(k/n) rPPM We have shown that the lower error rates of CPN can improve coding latency The Holevo capacity limit Alexander Holevo Steklov Mathematical Institute, Moscow, Russia Classical communication over a quantum channel Classical information transmission via quantum states The “channel” is determined by the receiver measurement itself. How do we know which receiver will get the highest capacity? xmtr quantum channel rcvr Quantum entropy bound Forney (1963), Gordon (1964), Holevo (1973), Yuen (1993) Channel capacity is the maximum of Holevo information (over prior probabilities, transmitter states, and POVMs) Holevo (1998), Schumacher, Westmoreland (1997) Giovannetti, Guha, Lloyd, Maccone, Shapiro, & Yuen (2004) 29 Limits of photon efficiency • Photon information efficiency (PIE) is the bits per photon • Achieving the Holevo limit will require optical codes and JDRs Holevo OOK + DD BPSK+Dolinar BPSK+homodyne Quantum polar codes can reach Holevo Wilde, Guha, arXiv: 1109.2591, IEEE 2012 Guha, Wilde, arXiv: 1202.0533, ISIT, 2012 DD limit also can be reached by BPSK with “Green Machine” JDR Guha, PRL (2011) Guha, Dutton, Shapiro arXiv:1102.1963v1 Proc. ISIT (2011) Shannon versus Holevo Capacity Limits Guha, PRL (2011) • Pure-loss bosonic channel (shown for BPSK alphabet) superadditive capacity Any projective measurement Any projective measurement Any projective measurement • With a joint quantum measurement, for any R<C∞, there is a code for which the optimal quantum measurement minimizes the Pr(error) of discriminating the codewords • Optimal measurement can be implemented as unitary transformation (beamsplitters, phase-shifters, squeezing, photon counting or Kerr nonlinearities) on the (optical) codeword followed by a sequence of projective measurements on the single-symbol state spaces Slicing JDR to achieve Holevo limit Performing the MPE measurement, treating the JDR (inner) code block as a waveform will allow us to reach the Helstrom limit 1 2 3 4 N N qubit unitary gate Circuit requires O(2N) single and two qubit gates [Solovay-Kitaev theorem] Can this circuit be simplified for codes that have certain symmetry properties? Conclusions and Ongoing research • We have demonstrated the first joint detection receiver – CPN reduces error rate of PPM demodulation (as compared to SQL) – This reduced error rate reduces coding latency (though does not increase capacity) • We have solved the long-standing problem of MPE discrimination of an arbitrary number of coherent states – This can be applied to reach the Holevo capacity by treating long codeword blocks as waveforms to be discrminated with a JDR – This is an interesting application of a small quantum computer! • The Holevo limit can be reached with coherent states – However, it requires new JDR receivers and optimal coding – Quantum polar codes can reach the Holevo limit – Still working on explicit optical implementations for the slicing or other optimal receivers Back-up The Binary Symmetric Channel: capacity • Binary symmetric channel: two inputs and two outputs – Used in early days to model the telegraph channel 0 0 B A 1 • Capacity – Capacity attained for equal prior probability on A – Binary entropy function 1 CPN vs. DD coding latency comparison • • • • • We took our experimental and ideal theory data for both DD and CPN systems to compare coding overhead and latency At ns=-5 dB (pulse energy Np = 1.25, M=4) we had Pe(DD) Peras(DD) Pe(CPN) Peras(CPN) Ideally 0.0 0.289 0.082 0.011 Experiment 0.004 0.287 0.092 0.052 We then calculated the RS block length required for coded error rate <10-10 At low rates, the higher number of erasures helps DD’s outer-coding, despite the higher uncoded error rate At higher rates (near capacity) the CPN greatly outperforms DD 105 104 DD nmin 103 CPN 102 101 RS rate (n/k)