Połączenie między generatorami stabilizatora a macierzami kontroli parzystości w kodzie Steane


15

Pracuję przez Mike'a i Ike (Nielsen i Chuang) do samokształcenia i czytam o kodach stabilizatorów w rozdziale 10. Jestem inżynierem elektrykiem z pewnym doświadczeniem w klasycznej teorii informacji, ale jestem w żadnym wypadku nie jest ekspertem w teorii kodowania algebraicznego. Moja abstrakcyjna algebra to w zasadzie tylko trochę więcej niż to, co znajduje się w dodatku.

Myślę, że całkowicie rozumiem konstrukcję Calderbank-Shor-Steane, w której do zbudowania kodu kwantowego stosuje się dwa klasyczne kody liniowe. Kod Steane'a jest konstruowany przy użyciu C1 (kod odwracania qbit) jako kodu Hamminga [7,4,3] oraz C2 (kod odwracania fazowego) jako tego samego kodu. Macierz kontroli parzystości kodu [7,4,3] to:

[000111101100111010101]
.

Generatory stabilizatorów dla kodu Steane'a można zapisać jako:

NameOperatorg1IIIXXXXg2IXXIIXXg3XIXIXIXg4IIIZZZZg5IZZIIZZg6ZIZIZIZ
gdzie ze względu na moje zdrowie psychiczneIIIXXXX=IIIXXXXi tak dalej.

W książce zaznaczono, że X i Z znajdują się w tych samych pozycjach, co 1 s w oryginalnym kodzie kontroli parzystości. Ćwiczenie 10.32 prosi o sprawdzenie, czy słowa kodowe dla kodu Steane są ustabilizowane przez ten zestaw. Oczywiście mógłbym to podłączyć i sprawdzić ręcznie. Stwierdzono jednak, że przy obserwacji podobieństw między macierzą kontroli parzystości a generatorem ćwiczenie jest „oczywiste”.

Widziałem ten fakt w innych miejscach ( http://www-bcf.usc.edu/~tbrun/Course/lecture21.pdf ), ale brakuje mi jakiejś (prawdopodobnie oczywistej) intuicji. Myślę, że brakuje mi jakiegoś dalszego połączenia klasycznych słów kodowych z kodami kwantowymi, innych niż sposób, w jaki są one używane do indeksowania podstawowych elementów w konstrukcji kodu (tj. Sekcja 10.4.2).

Odpowiedzi:


6

Jest tu kilka konwencji i intuicji, które być może pomogłyby przeliterować -

  • Bity znakowe kontra bity {0,1}

    Pierwszym krokiem jest uczynienie czegoś, co jest czasami nazywane „wielkim przesunięciem notacyjnym” i pomyślenie o bitach (nawet klasycznych) jako zakodowanych w znakach. Jest to produktywne, jeśli najbardziej interesuje Cię parzystość ciągów bitowych, ponieważ odwracanie bitów i odwracanie znaków w zasadzie działają w ten sam sposób. Mamy mapę 0+1 i 11 , tak że na przykład, sekwencja bitów (0,0,1,0,1) będzie reprezentowane przez sekwencję znaków (+1,+1,1,+1,1) .

    Pary parzystości bitów odpowiadają zatem iloczynom sekwencji znaków. Na przykład, tak jak uznalibyśmy 00101=0 jako obliczenie parzystości, możemy rozpoznać (+1)(+1)(1)(+1)(1)=+1 jako reprezentujący to samo obliczenie parzystości przy użyciu konwencji znaku.

    Ćwiczenie.Oblicz „parzystość” dla (1,1,+1,1) i dla (+1,1,+1,+1) . Czy to są te same?

  • Kontrole parzystości za pomocą bitów znaku

    W konwencji {0,1} -bitowej kontrole parzystości mają ładną reprezentację jako iloczyn iloczynu dwóch wektorów boolowskich, dzięki czemu możemy realizować skomplikowane obliczenia parzystości jako transformacje liniowe. Przechodząc na bity znakowe, nieuchronnie utraciliśmy połączenie z algebrą liniową na poziomie notacyjnym , ponieważ bierzemy produkty zamiast sum. Na poziomie obliczeniowym, ponieważ jest to tylko zmiana notacji, tak naprawdę nie musimy się zbytnio martwić. Ale na czysto matematycznym poziomie musimy teraz jeszcze raz przemyśleć, co robimy z macierzami kontroli parzystości.

    When we use sign bits, we may still represent a 'parity check matrix' as a matrix of 0s and 1s, instead of signs ±1. Why? One answer is that a row vector describing a parity check of bits is of a different type than the sequence of bits themselves: it describes a function on data, not the data itself. The array of 0s and 1s now just requires a different interpretation — instead of linear coefficients in a sum, they correspond to exponents in a product. If we have sign bits (s1,s2,,sn){1,+1}n, and we want to compute a parity check given by a row-vector (b1,b2,,bn){0,1}, the parity check is then computed by

    (s1)b1(s2)b2[](sn)bn{1,+1},
    where recall that s0=1 for all s. As with {0,1}-bits, you can think of the row (b1,b2,,bn) as just representing a 'mask' which determines which bits sj make a non-trivial contribution to the parity computation.

    Exercise. Compute the result of the parity check (0,1,0,1,0,1,0) on (+1,1,1,1,1,+1,1).

  • Eigenvalues as parities.

    The reason why we would want to encode bits in signs in quantum information theory is because of the way that information is stored in quantum states — or more to the point, the way that we can describe accessing that information. Specifically, we may talk a lot about the standard basis, but the reason why it is meaningful is because we can extract that information by measurement of an observable.

    This observable could just be the projector |11|, where |0 has eigenvalue 0 and |1 has eigenvalue 1, but it is often helpful to prefer to describe things in terms of the Pauli matrices. In this case, we would talk about the standard basis as the eigenbasis of the Z operator, in which case we have |0 as the +1-eigenvector of Z and |1 as the −1-eigenvector of Z.

    So: we have the emergence of sign-bits (in this case, eigenvalues) as representing the information stored in a qubit. And better still, we can do this in a way which is not specific to the standard basis: we can talk about information stored in the 'conjugate' basis, just by considering whether the state is an eigenstate of X, and what eigenvalue it has. But more than this, we can talk about the eigenvalues of a multi-qubit Pauli operator as encoding parities of multiple bits — the tensor product ZZ represents a way of accessing the product of the sign-bits, that is to say the parity, of two qubits in the standard basis. In this sense, the eigenvalue of a state with respect to a multi-qubit Pauli operator — if that eigenvalue is defined (i.e. in the case that the state is an eigenvalue of the Pauli operator) — is in effect the outcome of a parity calculation of information stored in some choice of basis for each of the qubits.

    |11ZZXX

    What is the parity of the state |+ with respect to XX? Does this state have a well-defined parity with respect to ZZ?

    Exercise. What is the parity of |Φ+=12(|00+|11) with respect to ZZ and XX?

  • Stabiliser generators as parity checks.

    We are now in a position to appreciate the role of stabiliser generators as being analogous to a parity check matrix. Consider the case of the 7-qubit CSS code, with generators

    GeneratorTensor factors1234567g1XXXXg2XXXXg3XXXXg4ZZZZg5ZZZZg6ZZZZ
    I've omitted the identity tensor factors above, as one might sometimes omit the 0s from a {0,1} matrix, and for the same reason: in a given stabiliser operator, the identity matrix corresponds to a tensor factor which is not included in the 'mask' of qubits for which we are computing the parity. For each generator, we are only interested in those tensor factors which are being acted on somehow, because those contribute to the parity outcome.

    Now, the 'codewords' (the encoded standard basis states) of the 7-qubit CSS code are given by

    |0L|0000000+|0001111+|0110011+|0111100+|1010101+|1011010+|1100110+|1101001=yC|y,|1L|1111111+|1110000+|1001100+|1000011+|0101010+|0100101+|0011001+|0010110=yC|y1111111,
    where C is the code generated by the bit-strings 0001111, 0110011, and 1010101. Notably, these bit-strings correspond to the positions of the X operators in the generators g1, g2, and g3. While those are stabilisers of the code (and represent parity checks as I've suggested above), we can also consider their action as operators which permute the standard basis. In particular, they will permute the elements of the code C, so that the terms involved in |0L and |1L will just be shuffled around.

    The generators g4, g5, and g6 above are all describing the parities of information encoded in standard basis states. The encoded basis states you are given are superpositions of codewords drawn from a linear code, and those codewords all have even parity with respect to the parity-check matrix from that code. As g4 through g6 just describe those same parity checks, it follows that the eigenvalue of the encoded basis states is +1 (corresponding to even parity).

    This is the way in which

    'with the observation about the similarities between the parity check matrix and the generator the exercise is "self evident"'

    — because the stabilisers either manifestly permute the standard basis terms in the two 'codewords', or manifestly are testing parity properties which by construction the codewords will have.

  • Moving beyond codewords

    The list of generators in the table you provide represent the first steps in a powerful technique, known as the stabiliser formalism, in which states are described using no more or less than the parity properties which are known to hold of them.

    Some states, such as standard basis states, conjugate basis states, and the perfectly entangled states |Φ+|00+|11 and |Ψ|01|10 can be completely characterised by their parity properties. (The state |Φ+ is the only one which is a +1-eigenvector of XX and ZZ; the state |Ψ is the only one which is a −1-eigenvector of both these operators.) These are known as stabiliser states, and one can consider how they are affected by unitary transformations and measurements by tracking how the parity properties themselves transform. For instance, a state which is stabilised by XX before applying a Hadamard on qubit 1, will be stabilised by ZX afterwards, because (HI)(XX)(HI)=ZX. Rather than transform the state, we transform the parity property which we know to hold of that state.

    You can use this also to characterise how subspaces characterised by these parity properties will transform. For instance, given an unknown state in the 7-qubit CSS code, I don't know enough about the state to tell you what state you will get if you apply Hadamards on all of the qubits, but I can tell you that it is stabilised by the generators gj=(H7)gj(H7), which consist of

    GeneratorTensor factors1234567g1ZZZZg2ZZZZg3ZZZZg4XXXXg5XXXXg6XXXX
    This is just a permutation of the generators of the 7-qubit CSS code, so I can conclude that the result is also a state in that same code.

    There is one thing about the stabiliser formalism which might seem mysterious at first: you aren't really dealing with information about the states that tells you anything about how they expand as superpositions of the standard basis. You're just dealing abstractly with the generators. And in fact, this is the point: you don't really want to spend your life writing out exponentially long superpositions all day, do you? What you really want are tools to allow you to reason about quantum states which require you to write things out as linear combinations as rarely as possible, because any time you write something as a linear combination, you are (a) making a lot of work for yourself, and (b) preferring some basis in a way which might prevent you from noticing some useful property which you can access using a different basis.

    Still: it is sometimes useful to reason about 'encoded states' in error correcting codes — for instance, in order to see what effect an operation such as H7 might have on the codespace of the 7-qubit code. What should one do instead of writing out superpositions?

    The answer is to describe these states in terms of observables — in terms of parity properties — to fix those states. For instance, just as |0 is the +1-eigenstate of Z, we can characterise the logical state |0L of the 7-qubit CSS code as the +1-eigenstate of

    ZL=ZZZZZZZ
    and similarly, |1L as the −1-eigenstate of ZL. (It is important that ZL=Z7 commutes with the generators {g1,,g6}, so that it is possible to be a +1-eigenstate of ZL at the same time as having the parity properties described by those generators.) This also allows us to move swiftly beyond the standard basis: using the fact that X7 anti commutes with Z7 the same way that X anti commutes with Z, and also as X7 commutes with the generators gi, we can describe |+L as being the +1-eigenstate of
    XL=XXXXXXX,
    and similarly, |L as the −1-eigenstate of XL. We may say that the encoded standard basis is, in particular, encoded in the parities of all of the qubits with respect to Z operators; and the encoded 'conjugate' basis is encoded in the parities of all of the qubits with respect to X operators.

    By fixing a notion of encoded operators, and using this to indirectly represent encoded states, we may observe that

    (H7)XL(H7)=ZL,(H7)ZL(H7)=XL,
    which is the same relation as obtains between X and Z with respect to conjugation by Hadamards; which allows us to conclude that for this encoding of information in the 7-qubit CSS code, H7 not only preserves the codespace but is an encoded Hadamard operation.

Thus we see that the idea of observables as a way of describing information about a quantum states in the form of sign bits — and in particular tensor products as a way of representing information about parities of bits — plays a central role in describing how the CSS code generators represent parity checks, and also in how we can describe properties of error correcting codes without reference to basis states.


After having read this I still don't get how it is obvious that X4X5X6X7 (I take this example) stabilize the code space. In your answer you seemed to use the fact you know what the code space looks like noticing that X4X5X6X7 applied on |0L gives |1L. What perturbs me is that if we talked about Z4Z5Z6Z7 operators, I would directly see the link with parity check matrix as their eigenvalues give the parity of the bits 4 to 7 exactly like the first line of parity check matrix. But the X operator are not diagonal in the 0/1 basis. So I don't get...
StarBucK

While it isn't ideal to fuss with the state-vectors for the code-words, from the expansion above we can see that X4X5X6X7 permutes the standard basis components of |0L, and similarly for the standard-basis components of |1L. While this picture involves non-trivial transformations of parts of the state, the overall effect is stabilisation. To see how to see the X stabilisers as parity-checks, the way you do with Z-stabilisers, maybe you could consider the effect of the X stabilisers of |+L and |L, and in the conjugate basis.
Niel de Beaudrap

Thank you for your answer. Ok so to be sure: do you agree that if we don't look at the basis of the code space but only at the parity check matrix and the generators, the only thing we can directly understand is the fact the Z generator can be read in the parity check matrix. But for the X generator even if appears they follow a similar structure it is not obvious to understand why without further calculation ? Because in the Nielsen&Chuang the way it is presented is as if it was obvious. So I wondered if I missed something ?
StarBucK

2

One way that you could construct what the codeword is is to project on the +1 eigenspace of the generators,

|C1=126(i=16(I+gi))|0000000.
Concentrate, to start with, one the first 3 generators
(I+g1)(I+g2)(I+g3).
If you expand this out, you'll see that it creates all the terms in the group (I,g1,g2,g3,g1g2,g1g3,g2g3,g1g2g3). Corresponding it to binary strings, the action of multiplying two terms (since X2=I) is just like addition modulo 2. So, contained within the code are all of the words generated by the parity check matrix (and this is a group, with group operation of addition modulo 2).

Now, if you multiply by one of the X stabilizers, that's like doing the addition modulo 2 on the corresponding bit strings. But, because we've already generated the group, by definition every group element is mapped to another (unique) group element. In other words, if I do

g1×{I,g1,g2,g3,g1g2,g1g3,g2g3,g1g2g3}={g1I,g1g2,g1g3,g2,g3,g1g2g3,g2g3},
I get back the set I started with (using g12=I, and the commutation of the stabilizers), and therefore I'm projecting onto the same state. Hence, the state is stabilized by g1 to g3.

You can effectively make the same argument for g4 to g6. I prefer to think about first applying a Hadamard to every qubit, so the Xs are changed to Zs and vice versa. The set of stabilizers are unchanged, so the code is unchanged, but the Z stabilizer is mapped to an X stabilizer, about which we have already argued.


1

What follows perhaps doesn't exactly answer your question, but instead aims to provide some background to help it become as 'self-evident' as your sources claim.

The Z operator has eigenstates |0 (with eigenvalue +1) and |1 (with eigenvalue 1). The ZZ operator on two qubits therefore has eigenstates

|00,|01,|10,|11
. The eigenvalues for these depend on the parity of the bit strings. For example, with |00 we multiply the +1 eigenvalues of the individual Z operators to get +1. For |11 we multiply the 1 eigenvalues together and also get +1 for ZZ. So both these even parity bit strings have eigenvalue +1, as does any superposition of them. For both odd parity states (|01 and |10) we must multiply a +1 with a 1, and get a 1 eigenvalue for ZZ.

Note also that superpositions of bit strings with fixed parity (such as some α|00+β|00) are also eigenstates, and have the eigenvalue associated with their parity. So measuring ZZ would not collapse such a superposition.

This analysis remains valid as we go to higher number of qubits. So if we want to know about the parity of qubits 1, 3, 5, and 7 (to pick a pertinent example), we could use the operator ZIZIZIZ. If we measure this and get the outcome +1, we know that this subset of qubits has a state represented by an even parity bit string, or a superposition thereof. If we get 1, we know that it is an odd parity state.

This allows us to write the [7,4,3] Hamming code using the notation of quantum stabilizer codes. Each parity check is turned into a stabilizer generator which has I on every qubit not involved in the check, and Z on every qubit that is. The resulting code will protect against errors that anticommute with Z (and therefore have the effect of flipping bits).

Of course, qubits do not restrict us to only working in the Z basis. We could encode our qubits for a classical Hamming code in the |+ and | states instead. These are the eigenstates of X, so you just need to replace Z with X in everything I've said to see how this kind of code works. It would protect against errors that anticommute with X (and so have the effect of flipping phase).

The magic of the Steane code, of course, is that it does both at the same time and protects against everything.

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