Optical Communications at the Quantum Limit

Zachary DuttonRaytheon BBN Technologies Cambridge, MA

ICQIQC 2013IIScBangolore, India11 January, 2013

Quantum Information Processing (QuIP group)

Physical Technologies

(e.g. superconducting qubits, superconducting photon

detectors, SPDC sources)

Applications(e.g. Entanglement enhanced LADAR; secret key distribution, quantum computation)

Devices(e.g. quantum

repeater stations)

System tools(e.g. quantum

functional language and compilation )

Components(e.g. multi-qubit

chips, entanglement

sources)

Physical Theory(e.g. non-standard optical receivers)

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

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

• Two 10 mk dilution refrigerators

• Research on nonlinear optical effects at microwave frequencies

– Coherent population trapping (CPT)

Surface code structure

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

K. Murali, et. al PRL 93, 087003 (2004)W. R. Kelly, et. al, PRL 104 163601 (2010)

3-qubit / 2-cavity sub-cell

E. Magesan, et. al PRL 109, 080505 (2012)

Light is very good at carrying information– Much higher bandwidth (higher data rates) than RF– Long range (suitable for deep space communication)

– What are the limits of optical communication and how can we reach them?

– It requires quantum theory to model a physical channel

Holevo limitHomodyneHeterodyne

Our “PIECOMM” team

• Saikat Guha (PI)

• Zachary Dutton

• Hari Krovi

• Monika Patel

• Jeff Chen

• Jonathan Habif

• Richard Lazarus

• Jeffrey H. Shapiro– Nivedita Chandrasekaran

• Seth Lloyd• Lizhong Zheng

– Hye Won Chung• Gregory Wornell

– Yuval Kochman– Ligong Wang

• Franco N. C. Wong– Valentina Schettini

• Karl Berggren– Francesco Bellei– Hasan Korre

BBN MIT

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

Re

Im

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

• Binary phase shift keying (BPSK) modulation

• Quadrature phase shift keying (QPSK) modulation

Direct detection

Homodyne detection

Heterodyne detection

Re

Im

Re

Im

Re

Im

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 , with Prob. and– MPE measurement: Minimize

such that– Helstrom measurement

– 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

• 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)

Cook, Martin, Geremia Nature (2007)

Dolinar (1976)

Becerra, et. al, Nature Phonics (in press)

The “Generalized Kennedy” Receiver“off”

“on”

click

no click

say “off”

say “on”

Tsujino, PRL (2011) arXiv:1103.5592

DDGKHelstrom

Pe (OOK)

Np(dB)

• Utilizes optimized (but constant) nulling• Recently demonstrated by NICT and unambiguously beats SQL

Exactnulling

PPM demodulation using the Conditional Pulse Nulling (CPN) receiver

1Null pulse-1 and DD 2

Null pulse-2 and DD

Null pulse-3 and DD

1DD pulse 2, 3, 41

32

1k k

22

DD pulse 3, 4 kk

43

33

DD pulse 4 44

SPD

DecisionNulling

PPM PulsesDecoder

Dolinar, MIT Ph.D. Thesis 1976, TDA Progress Report, 42-72, 1982Guha, Habif, Takeoka, J. Mod. Optics, Vol. 58, Nos. 3–4, 10–20, 257–265, 2011

Nulling “Decision Tree”

Reaches sub-SQL error demodulation of a codeword of multiple symbols (joint detection receiver – JDR)

System Diagram of a CPN Receiver

Chen, Habif, Dutton, Lazarus & Guha, Nature Photonics (2012), Xiv:1111.4017

Experimental results

DDCPNHelstrom

Dashed: idealSolid: model

• 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

a b

pNa

TDTT

q

q

qbi

null

ip

eN

eNd

)1(

P

PPT

P

Nddd

NNN

)]122()cos1)(11[(

])cos1(2[|| 2

0

DDD

q

qba

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– 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)

Nair & Guha, arXiv:1212.2048

The slicing receiver

U is a highly non-linear operation

Infinite dimensional coherent states can be “compressed” into finite dimensional qubits

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

n = 2, 10, 30, 100 slices n = 2, 10, 30, 100 slices

The receiver is seen here to work for 2 and 3 state discrimination and is fully generalizable

• The final step is then a projective measurement of the register

• Note that this is essentially a small continuous variable optical quantum computer

Fundamental limits of capacity

Connecting of sub-SQL to capacity

Giovannetti, Guha, Lloyd, Maccone, & Shapiro (2004)

Shapiro, Guha, Erkmen (2005)

Guha, PRL (2011)

• 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

Reliable communication over a noisy channel

Claude Shannon

“Father of information theory”1916-2001

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

Classical information transmission via classical symbols

Shannon (1948)

noisy channel

Classical communication over a classical channel

• 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

Shannon’s intuition

[100100101]

[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.

(decoding)

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

• 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

(N,R) outer encoder

(n,r) inner encoder

Modu-lator

Physical channel

OpticalReceiver

Demodu-lator

Detec-tor

Inner decoder

Outer decoder

Joint-detection receiver (JDR)

inner super-channel: Shannon capacity: Cn > C1

Guha, Dutton, ShapiroarXiv:1102.1963v1 [quant-ph]Proc. ISIT 2011

The Holevo capacity limit

Alexander Holevo

Steklov Mathematical Institute,Moscow, Russia

29

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?

Channel capacity is the maximum of Holevo information (over prior probabilities, transmitter states, and POVMs)

Quantum entropy bound

Holevo (1998), Schumacher, Westmoreland (1997)

quantum channelxmtr rcvr

Forney (1963), Gordon (1964), Holevo (1973), Yuen (1993)

Classical communication over a quantum channel

Giovannetti, Guha, Lloyd, Maccone, Shapiro, & Yuen (2004)

Limits of photon efficiency• Photon information efficiency (PIE) is the bits per photon • Achieving the Holevo limit will require optical codes and JDRs

HolevoOOK + DDBPSK+DolinarBPSK+homodyne

Quantum polar codes can reach HolevoWilde, Guha, arXiv: 1109.2591, IEEE 2012Guha, 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

• Pure-loss bosonic channel (shown for BPSK alphabet)

• 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

superadditive capacity

Guha, PRL (2011)

• 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

Any projective measurementAny projective measurement

Any projective measurement

Slicing JDR to achieve Holevo limit

1 2 3 4 N

N qubit unitary gateCircuit requires O(2N) single and two qubit gates [Solovay-Kitaev theorem]Can this circuit be simplified for codes that have certain symmetry properties?

Performing the MPE measurement, treating the JDR (inner) code block as a waveform will allow us to reach the Helstrom limit

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

• Capacity– Capacity attained for equal prior probability on A– Binary entropy function

A B

0

1

0

1

CPN vs. DD coding latency comparison

RS rate (n/k)

105

104

103

102

101

nmin

DD

CPN

• 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

• 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

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