Belief propagation decoding of quantum channels by passing quantum messages

Transcription

1 Belief propagation decoding of quantum channels by passing quantum messages arxiv: QIP 27 Joseph M. Renes

2 To do research in quantum information theory, pick a favorite text on classical information theory, open to a chapter, and translate the contents into quantum-mechanical language. Based on belief propagation decoding IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 59, NO. 2, DECEMBER Spatially Coupled Ensembles Universally Achieve Capacity Under Belief Propagation Shrinivas Kudekar, Tom Richardson, Fellow,IEEE,and RüdigerL.Urbanke arxiv: Abstract e investigate spatially coupled code ensembles. For transmission over the binary erasure channel, it was recently shown that spatial coupling increases the belief propagation threshold of the ensemble to essentially the maximum a priori threshold of the underlying component ensemble. This explains why convolutional LDPC ensembles, originally introduced by Felström and Zigangirov, perform so well over this channel. e show that the equivalent result holds true for transmission over general binary-input memoryless output-symmetric channels. More precisely, given a desired error probability and a gap to capacity, we can construct a spatially coupled ensemble that fulfills these constraints universally on this class of channels under belief propagation decoding. In fact, most codes in this ensemble have this property. The quantifier universal refers to the single ensemble/code that is good for all channels but we assume that the channel is known at the receiver. The key technical result is a proof that, under belief-propagation decoding, spatially coupled ensembles achieve essentially the area threshold of the underlying In the first 5 years, coding theory focused on the construction of algebraic coding schemes and algorithms that were capable of exploiting the algebraic structure. Two early highlights of this line of research were the introduction of the Bose Chaudhuri Hocquenghem (BCH) codes [5], [6] as well as the Reed Solomon (RS) codes [7]. Berlekamp devised an efficient decoding algorithm [8], andthisalgorithmwasthen interpreted by Massey as an algorithm for finding the shortest feedback-shift register that generates a given sequence [9]. More recently, Sudan introduced a listdecoding algorithm forrs codes that decodes beyond the guaranteed error-correcting radius []. Guruswami and Sudan improved upon this algorithm [] and Koetter and Vardy showed how to handle soft information [2]. Another important branch started with the introduction of convolutional codes [3] by Elias and the introduction of the Quantum belief propagation decoding?

3 2 3 4 (3.9) í î ì ó ó ó ó ó ó ó ó ó = H. ó ó ó ó ó ó ó ó ó ï Belief propagation: message passing algorithm for performing inference in a graphical model øe bipartite graph representing C(H) is shown on the le of Figure 3.. Each check decoder infers channel input from output; code can be described by a graphical model early precursor: Bethe-Peierls approximation in statistical physics Figure 3.: Le : Tanner graph of H given in (3.9). Right: Tanner graph of [7, 4, 3] Hamming code corresponding to the parity-check matrix on page 5. øis graph is discussed in Example 3.. node represents one linear constraint (one row of H). For the particular example we start with 2 degrees of freedom (2 variable nodes). øe constraints reduce the number of degrees of freedom by at most (and exactly by if all these constraints are linearly independent as in this speci c example). øerefore at least degrees of freedom remain. It follows that the shown code has rate (at least) one-half. n many applications in statistics and machine learning besides coding: inference, optimization, constraint satisfaction 3.4. Low-Density Parity-Check Codes In a nutshell, low-density parity-check (LDPC) codes are linear codes that have at least one sparse Tanner graph. øe primary reason for focusing on such codes is

4 Quantum belief propagation decoding? Can use classical algorithm for usual stabilizer decoding Optimal for: But not, e.g. amplitude damping Let s consider CQ channels for simplicity Need to infer channel input from quantum output (not trying to compute marginals of quantum states)

5 Results arxiv: BP decoder for pure state channel, tree codes Also works for polar codes: Efficient, capacity-achieving decoder for BPSK over lossy Bosonic channel And for quantum communication, part of conjugate basis decoder: Efficient, capacity-achieving decoder for amplitude damping

6 . Coding setup 2. Factor graphs 3. Classical BP 4. Quantum BP 5. Many open questions National Park Service. Coding setup 2. Factor graphs 3. Classical BP 4. Quantum BP 5. Many open questions Outline

7 Coding setup Shannon scenario: stochastic iid noise. not adversarial; fault-free decoding î Linear code( message encoder decoder decoded message bitwise decoding of random codeword classically, marginalize joint distribution P X n Y n =yn! P X i Y n =yn quantumly, perform Helstrom measurement for each bit not X n B n! X i B n

8 Factor graphs factorizeable joint probability f g h P (x,,, )= Z f(x, )g( )h(,, ) x x2 x3 x4 P (x )= easier to compute marginals: X X P (x,,, )= Z f(x, )g( ),, X, h(,, ) (y x ) (y 2 ) (y 3 ) (y 4 ) x [x + =] [x + + =] P X n Y n (x n,y n )= C channel input & output distribution: [x n 2 C] ny (y j x j ). j=

9 Classical BP decoding y symmetric binary input channel: only care about likelihood ratio `(y) = (y ) (y ) (y x ) (y 2 ) x [x + =] x i Ñ y n is a channel; want `py n q (y 3 ) (y 4 ) [x + + =] assume tree factor graph for each node, associate a channel to all its leaves x BP recursively computes starting from the leaves `py n q

10 Classical BP decoding y symmetric binary input channel: only care about likelihood ratio `(y) = (y ) (y ) (y x ) (y 2 ) x [x + =] x i Ñ y n is a channel; want `py n q (y 3 ) (y 4 ) [x + + =] assume tree factor graph for each node, associate a channel to all its leaves x BP recursively computes starting from the leaves `py n q

11 Classical BP decoding y symmetric binary input channel: only care about likelihood ratio `(y) = (y ) (y ) (y x ) (y 2 ) x [x + =] x i Ñ y n is a channel; want `py n q (y 3 ) (y 4 ) [x + + =] assume tree factor graph for each node, associate a channel to all its leaves x BP recursively computes starting from the leaves `py n q

12 Classical BP decoding y symmetric binary input channel: only care about likelihood ratio `(y) = (y ) (y ) (y x ) (y 2 ) x [x + =] x i Ñ y n is a channel; want `py n q (y 3 ) (y 4 ) [x + + =] assume tree factor graph for each node, associate a channel to all its leaves x BP recursively computes starting from the leaves `py n q

13 Classical BP decoding BP rules specify two kinds of channel convolution f b b Variable node x f {2p b ` b q {2p b ` b q Check node BP finds exact marginals on trees Can run algorithm for all codeword bits concurrently Also works on loopy LDPC factor graphs

14 Barefoot Jake

15 Quantum BP decoding Now the outputs are quantum, no likelihood function ant to perform Helstrom measurement x Can we combine Helstrom measurements along the branches? Can we repackage the outputs as with likelihood? Try the simplest case: pure state outputs ` y y x` y cos optimal measurement is σx

16 Quantum BP decoding x` y cos ` y y x f b b f {2p b ` b q {2p b ` b q convolution yields pure states: repackage in a single qubit with U f p, q ` yb ` y yb y f y cos f cos cos convolution gives a heralded mixture of pure states! unitary U f gives ÿ jpt,u p j jyxj b f j yx f j p 2 p ` cos cos q cos f cos ` cos cos f ` cos cos cos cos cos cos

17 Quantum BP decoding f y needs to know input qubit angles swap U f U f H U f Helstrom measurement x ÿ jpt,u p j f j yx f j b jyxj f pass qubits and some classical bits: sufficient statistic decode all codeword bits sequentially, unwinding each time O(n 2 ) implementation of all Helstrom measurements

18 Quantum BP & polar codes Variable and check convolutions = better and worse synth channels Polar decoder has tree structure (for message not codeword bits, tho) PRL 9, 554 (22) P H Y S I C A L R E V I E L E T T E R S Efficient Polar Coding of Quantum Information Joseph M. Renes, Frédéric Dupuis, and Renato Renner Institut für Theoretische Physik, ETH Zurich, CH-893 Zürich, Switzerland (Received 23 March 22; published August 22) Polar coding, introduced 28 by Arkan, is the first (very) efficiently encodable and decodable coding scheme whose information transmission rate provably achieves the Shannon bound for classical discrete memoryless channels in the asymptotic limit of large block sizes. Here, we study the use of polar codes for the transmission of quantum information. Focusing on the case of qubit Pauli channels and qubit erasure s, we use classical polar codes to construct a coding scheme that asymptotically achieves a net coherent information using efficient encoding and decoding operations and hared entanglement between sender and receiver, but ate of preshared entanglement 39 IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 6, NO. 6, JUNE 24 Polar Codes for Private and Quantum Communication Over Arbitrary Channels Joseph M. Renes, Member, IEEE, andmarkm.ilde,senior Member, IEEE Abstract e constructnewpolarcodingschemesforthe transmissionofquantum or private classical information over arbitrary quantum channels. In the former case, our coding scheme achieves the symmetriccoherentinformation, and in the metric private information. Both schemesarebuilt capable of transmitting classical merging two week ending 3 AUGUST 22 channel uses. These codes exploit the channel polarization effect whereby a particular recursive encoding induces a set of virtual channels, such that some of the virtual channels are perfect for data transmission while the others are useless for this task. The fraction containing perfect virtual channels is equal to the channel s symmetric capacity. this paper, we offer new polar coding schemes for rmation or for privately transmitting d on ideas of Quantum polar decoder uses polar decoder for classical amplitude and phase info : amplitude = classical Z channel, phase = pure state channel O(n 2 ) decoder for capacity-achieving quantum polar code for amplitude damping

19 Open Questions oldsloat.blogspot

20 Open other channels? e.g. classical coding for amplitude damping? loops? spatially-coupled LDPC codes? use density matrix BP? need sufficient statistics, local tree to global relation to tensor networks? junction-tree algorithm? Viterbi decoder (blockwise decoder)? fully quantum version? other tasks besides decoding? Questions oldsloat.blogspot