Compared to 2 , the input x is uniformly random and not always zero, but it is given at the channel output. The form of the check and variable convolutions also applies to channels with quantum output 2. We need only replace the probability distributions over the output alphabet by quantum states. Abusing notation, let us denote by W x the quantum state of the output of W given input x. This includes the previous case by considering commuting W x. The the variable and check node convolutions are now just. To properly generalize the BP decoding algorithm we need a 'sufficient statistic' for the quantum channels at the various nodes.

For binary-input pure state channels, it turns out that a combination of classical bits and just one qubit suffices. The channel outputs can always be represented by a qubit, so suppose that W outputs , where. Note that the overlap of the two states is and the Helstrom measurement for these two states is measurement of the operator. The convolution outputs either or , which are again two pure states, with an overlap angle given by. The following unitary transformation compresses the states to the first qubit, leaving the second in the state :. To combine more than two channels, we just perform the pairwise convolution sequentially.

Thus, the convolution of pure state channels can itself be represented as a pure state channel. The convolution is more complicated because the outputs are no longer pure. However, applying the unitary results in a CQ state of the form We are free to measure the second qubit, and conditional state of the first qubit is again one of two pure states, though now the overlap depends on the measurement outcome j.

In particular, , , and the two overlaps are given by. Therefore, the convolution of pure state channels can be represented by two pure state channels, corresponding to the two measurement outcomes.

As before, several channels can be combined sequentially. The quantum decoding algorithm now proceeds as in classical BP, taking the quantum outputs of the channels and combining them at variable and check nodes.

## Liebler : Implementing gradient descent decoding

At a variable node the algorithm combines the outputs using and forwards the output to its parent node. At check nodes the algorithm applies , measures the second qubit, and forwards both the qubit and the measurement result to its parent node. The classical messages are required to inform parent variable nodes how to choose the angles in subsequent unitaries.

Ultimately this procedure results in one qubit at the root node such that measurement of corresponds to the optimal Helstrom measurement for the associated bit. This then is sufficient to estimate one input bit. Starting at the leaves, the outputs of all but the first channel can be immediately passed to their corresponding variable nodes, since these variable nodes do not have any other outward branches.

Formally this follows from the convolution rules by considering convolution with a trivial channel, having. The output of the first channel, meanwhile, must wait to be combined according to the convolution with several other qubit messages. Next, since 2 and 4 are connected by a check node, we combine qubits 2 and 4 into one qubit 2 and one classical bit 4 by applying and measuring the 4th qubit.

As qubits 1 and 3 are connected by a variable node, we can simultaneously combine these with. Finally, we combine qubits 1 and 2 by applying , where and , depending on the value j of the earlier measurement. A quantum circuit implementing these steps is shown in figure 2. Figure 2. Circuit decoding the first bit of the code depicted in figure 1.

## Fundamentals Of Codes, Graphs, And Iterative Decoding

The first convolution is the second for and , depending on the value j of the measurement outcome in the bottom wire. The symbol denotes that the qubit is discarded. The final Hadamard gate and measurement implement the Helstrom measurement. One drawback is that the above procedure implements the Helstrom measurement destructively, since once we estimate the first bit we no longer have the original channel output in order to estimate the second bit. And we cannot run the algorithm backwards to reproduce the channel output as we have made measurements at every check node.

To implement the Helstrom measurement as non-destructively as possible, we can leave the CQ output states unmeasured and instead use the classical subsystems to coherently control the variable node unitaries. In this way the steps in the algorithm can be reversed, save the final measurement. For example, in figure 2 all output qubits are kept and the classical measurement and subsequent conditioning of the second gate is performed by a coherent conditional gate involving three qubits.

Denoting the unitary action of the algorithm for the j th bit by V j , the Helstrom measurement can be implemented by the projective measurement with projectors , where denotes the k th basis projector on the j th qubit. Each V j is composed of O n gates, yielding an overall circuit size of to decode all bits. Supposing that the code is designed such that the j th input bit can be estimated with error no larger than , Gao's non-commutative union bound [ 25 ] implies that the error in sequentially estimating all bits is no worse than.

Polar codes for the pure state channel may also be decoded with this algorithm. The difference is that successive cancellation does not use the factor graph of the code, but a graph related to a fixed reversible encoding circuit.

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Importantly, the graph associated to each input of the encoding circuit is a tree. In fact, each such graph has logarithmic depth from all channel factors to each variable, and every node has degree three. Unlike the BP decoder, however, the successive cancellation decoder used by polar codes takes previously decoded bits into account.

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But these bits can be handled by the BP decoder since the pure state channel is symmetric in the manner described at the end of section 2. There, the value of the previous bits is incorporated into the better channel by appropriately permuting the output symbols, which is equivalent to flipping the input value. Similarly, for the pure state channel, applying to the output is equivalent to flipping the input. Therefore, the quantum BP decoding algorithm gives a successive cancellation decoder for polar codes over the pure loss Bosonic channel using the BPSK constellation [ 21 ].

The idea behind the quantum polar coding scheme of [ 16 , 18 ] is to decompose the problem of transmitting quantum information over a channel into transmitting classical information about two conjugate observables, 'amplitude' and 'phase', consider polar codes for each subproblem, and then combine the coding schemes using CSS codes at the encoder and coherent sequential decoding of amplitude and phase at the decoder.

This decoding strategy is depicted in [ 16 ], figure 3 for Pauli channels and [ 26 , figure 1] for the general case. As detailed in [ 18 ], the two classical transmission tasks are to transmit 'amplitude' information over the CQ channel given by and 'phase' information over the CQ channel given by.

Here is an arbitrary basis, and we choose that of for convenience, while is a bipartite pure state in this same basis with coefficients of our choosing. See [ 18 ] for the precise relation to the conjugate observables and. Let us now show how to build a decoder for the amplitude damping channel with damping parameter. Therefore we can use the classical polar encoder and decoder for this channel [ 27 ]. Since the Z channel is not symmetric, the optimal input distribution in the capacity formula is not the uniform distribution, but one with probabilities p and.

Now suppose that the bipartite pure state in the phase channel is the state. Abusing notation slightly and denoting the channel outputs , it is not difficult to verify that for ,. Each of these states is a CQ state with the first qubit pure and the second qubit classical, just as in a output.

## Belief propagation decoding of quantum channels by passing quantum messages

Hence the decoder can begin just as at a step, measuring the second qubit to determine the angle associated to the first qubit. The rate achievable by the quantum polar code construction is simply , where and. A cumbersome but straightforward calculation confirms that R equals the capacity of the channel, , for h 2 is the binary entropy [ 28 , proposition Moreover, since the amplitude damping channel is degradable, the arguments in [ 16 ] ensure that no entanglement-assistance is required to meet the CSS constraint when constructing the quantum polar code.

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We have presented a BP algorithm for bitwise decoding of CQ channels which operates by passing quantum messages on tree factor graphs, and shown several applications to polar codes. This invites the study of quantum message passing algorithms, and not just in the context of decoding. More generally we may look for BP and related algorithms for any task of statistical inference where the input data comes in the form of many quantum bits, for instance in quantum metrology. This work also raises many interesting questions.

Most immediately in the context of decoding is whether the complexity of the algorithm can be reduced for structured factor graphs. Classical polar codes, for instance, have decoding complexity. Can this also be achieved for the pure state channel? Similarly, can one find a quantum version of the max-product or Viterbi algorithm for determining the most likely x 1 n given the channel outputs?

### The purpose of this course

More generally, it would be very interesting to understand how to run the algorithm on a factor graph with loops, or how it can be modified to handle some set of non-pure output states. In the former case it may be useful to explore the characterization of loopy BP as a variational problem [ 1 , 29 ]. Perhaps in the latter case one can make use of the work on quantum sufficiency see e. Another interesting question with potentially far-reaching consequences is the relation of the BP algorithm to tensor network methods.

The problem of marginalization in the commutative setting is explicitly treated as tensor network contraction in [ 14 ], and the particulars of the quantum BP decoder bear a similarity with the data gathering approach using tensor network states in [ 32 ]. Can the methods of approximating quantum states by tensor networks be used to create efficient approximate decoders?