Nonlocal quantum information in bipartite quantum error correction
QQuantum Information Processing manuscript No. (will be inserted by the editor)
Nonlocal quantum information in bipartite quantum error correction
Mark M. Wilde · David Fattal
Received: November 2, 2018/ Accepted:
Abstract
We show how to convert an arbitrary stabilizer code into a bipartite quantum code. A bipartitequantum code is one that involves two senders and one receiver. The two senders exploit both nonlocal andlocal quantum resources to encode quantum information with local encoding circuits. They transmit theirencoded quantum data to a single receiver who then decodes the transmitted quantum information. Thenonlocal resources in a bipartite code are ebits and nonlocal information qubits and the local resourcesare ancillas and local information qubits. The technique of bipartite quantum error correction is useful inboth the quantum communication scenario described above and in fault-tolerant quantum computation.It has application in fault-tolerant quantum computation because we can prepare nonlocal resourcesoffline and exploit local encoding circuits. In particular, we derive an encoding circuit for a bipartiteversion of the Steane code that is local and additionally requires only nearest-neighbor interactions. Wehave simulated this encoding in the CNOT extended rectangle with a publicly available fault-tolerantsimulation software. The result is that there is an improvement in the “pseudothreshold” with respect tothe baseline Steane code, under the assumption that quantum memory errors occur less frequently thanquantum gate errors.
PACS · Quantum error correction is the theory upon which future quantum computation and quantum commu-nication devices will depend for reliable operation [43,24,8,23]. This theory is the formal “quantization”of the classical error correction theory [37] and borrows several fundamental concepts from the classicaltheory, such as digitization, redundancy, and measurement of errors. Though, the way in which theseconcepts manifest in quantum error correction is different from the way that they manifest in classicalerror correction.The exploitation of several different forms of “quantum redundancy” [9] has been a crucial componentof progress [43,35,4,2,41,7,6,34,52] in the theory of quantum error correction. First, Shor realized thatwe can obtain the quantum redundancy necessary for a quantum error-correcting code by entanglinginformation qubits with extra “ancilla” qubits [43]. Here, the ancilla qubits play the role of the resourcefor quantum redundancy. Ten years later, Kribs et al. realized that a noisy qubit, a so-called gauge qubit,is useful for quantum redundancy and formulated the theory of operator quantum error correction (alsoknown as subsystem quantum error correction) [35]. Other researchers then showed that quantum codeswith gauge qubits improve the noise threshold for reliable fault-tolerant quantum computation [4,2,41].Shortly after the Kribs et al . result, Brun et al. realized that an ebit , a bipartite maximally entangled Bellstate shared between a sender and receiver, is useful for quantum redundancy, and they formulated the
M.M.W. acknowledges support from the MDEIE (Qu´ebec) PSR-SIIRI international collaboration grant.Mark M. Wilde is a postdoctoral fellow with the School of Computer Science, McGill University, Montreal, Quebec,Canada H3A 2A7 E-mail: [email protected] · David Fattal is with the Information and Quantum Systems Laboratory,Hewlett-Packard Laboratories, 1501 Page Mill Road, MS 1123, Palo Alto, California, 94304-1100, USA a r X i v : . [ qu a n t - ph ] D ec heory of entanglement-assisted quantum error correction [7,6]. Kremsky et al. followed by showing howancillas and ebits are useful for the simultaneous transmission of classical and quantum information andformulated the classically-enhanced theory of quantum error correction [34,52]. These latter two theoriesof quantum error correction emerged from advances in quantum Shannon theory, in particular, from thefather protocol [16,15] and the classically-enhanced quantum communication protocol [17], respectively.In this paper, we develop a version of the stabilizer formalism for quantum error correction that wename the bipartite stabilizer formalism . A bipartite quantum error-correcting code is useful in a quan-tum communication scenario in which two senders encode quantum information by exploiting nonlocalresources that they share. They both then transmit this encoded quantum information to a single receiverwho decodes the transmitted information.We also introduce a new form of quantum redundancy to the theory of quantum error correction: a nonlocal information qubit . The standard form of a nonlocal information qubit is as follows: | ϕ (cid:105) AB ≡ α | (cid:105) AB + β | (cid:105) AB , where the coefficients α and β are arbitrary complex coefficients such that | α | + | β | = 1 and thesuperscripts indicate that one party, Alice, possesses the first qubit and another party, Bob, possessesthe second qubit. We discuss this resource in more detail later, but suffice it for now to say that itspower stands somewhere in between an information qubit (or logical qubit) and an ebit. In this sense,it is perhaps most similar to a coherent bit channel [28,54,53]. Nonlocal information qubits may arisenaturally in the setting of quantum network protocols [36,33] or distributed quantum computation [27,10,21,20,39]. In the sections of this paper on quantum communication, we simply assume that twosenders have such a resource available, perhaps distributed to them from some “source” party, and thesenders would like to exploit it for quantum communication purposes.We also show how an example of a bipartite quantum code, a variant of the Steane code, can leadto increased performance in fault-tolerant quantum computation (under certain assumptions) [44,3,23].The increased performance occurs for this code because its encoding circuit is localized and has fewererror locations than the encoding circuit for the baseline Steane code. Additionally, the code retainsthe error-correcting properties of the stabilizer code from which it is derived, ensuring the ability tocorrect errors on all qubits (similar to the example in Ref. [42]). In particular, we present a version ofthe Steane code [40] that requires only nearest neighbor interactions among four qubits for its encoding,assuming that the seven-qubit quantum register is initialized with three ebits and one information qubit.Simulations then demonstrate that this version of the Steane code outperforms the baseline Steane code,under the assumption that quantum memory errors occur less frequently than quantum gate errors.The present work represents an extension of the entanglement-assisted stabilizer formalism. We ex-ploit recent ideas from the structure of entanglement-assisted quantum codes [50,48] and entanglementmeasures of stabilizer states [22] to construct our bipartite quantum codes. We motivate new ideas forresearch pursuits in network quantum Shannon theory [57], specifically related to the multiple accessquantum channel [57,56,1,32,59,31].The next section of this paper develops the resource of a nonlocal information qubit. Section 3 showshow to construct a two-sender one-receiver quantum code from any stabilizer code, under the assumptionthat the two senders possess ebits and nonlocal information qubits. That section also includes an exampleof this type of code. We then present our fault-tolerant quantum computation simulation results for theperformance of the CNOT extended rectangle [3] using a bipartite variation of the Steane code [40]. Wefinally conclude with some open questions for investigation. | ϕ (cid:105) AB ≡ α | (cid:105) AB + β | (cid:105) AB , (1) Information theorists would typically denote such a scenario as a “multiple access coding” scenario [11]. This name istypically reserved for more exotic coding structures such as those used in code division multiple access [47] or superpositioncoding [60]. We choose the name “bipartite quantum error correction” to distinguish the coding structure presented herebecause it is not as exotic as either of the former methods (it simply cuts a stabilizer code into two parts). Z A Z B stabilizes the nonlocal qubit because the state in (1) is in the+1-eigenspace of the operator Z A Z B . Local errors of the form X A or X B anticommute with the operator Z A Z B . Claire can detect these errors by measuring the operator Z A Z B if she possesses both qubits afterAlice and Bob transmit them to her. It is also possible to manipulate the quantum information in thenonlocal information qubit through the use of logical operators. The X logical operator of the nonlocalinformation qubit is X A X B and the Z logical operator is either Z A or Z B . The logical Hamadardoperation is a nonlocal operation that transforms | (cid:105) AB to ( | (cid:105) AB + | (cid:105) AB ) / √ | (cid:105) AB to ( | (cid:105) AB −| (cid:105) AB ) / √
2. One cannot fully manipulate the quantum information in the nonlocal information qubitunless one possesses both qubits or unless, in some cases, we allow for classical communication betweenboth senders so that, e.g., they can apply a coordinated X rotation to implement the logical X operator.We can represent the stabilizer operator ZZ of a nonlocal information qubit as the following binaryvector by exploiting the Pauli-to-binary isomorphism (See Ref. [24]): H ≡ (cid:2) (cid:12)(cid:12) (cid:3) ≡ (cid:2) H AZ H BZ (cid:12)(cid:12) H AX H BX (cid:3) , where the matrix to the left of the vertical bar captures the “Z” part of the operator and the matrix tothe right of the vertical bar captures the “X” part of the operator. We can represent the logical operators XX and ZI with the following two respective row vectors: L ≡ (cid:20) (cid:12)(cid:12)(cid:12)(cid:12) (cid:21) ≡ (cid:2) L AZ L BZ (cid:12)(cid:12) L AX L BX (cid:3) . Let H A denote the following matrix: H A ≡ (cid:2) H AZ (cid:12)(cid:12) H AX (cid:3) , (2)and let H B , L A , and L B denote similarly defined matrices. Let G denote the following matrix: G ≡ (cid:20) HL (cid:21) , and we can define G A and G B similarly to the definitions of H A , H B , L A , and L B . For any given matrixof the form F ≡ [ F Z | F X ], we can define a corresponding symplectic product matrix Ω F where Ω F ≡ F Z F TX + F X F TZ , (3)and addition is binary. It is straightforward to check that the following relations hold for the nonlocalinformation qubit: 12 rank ( Ω L ) = 12 rank ( Ω L A ) = 12 rank ( Ω L B )= 12 rank ( Ω G )= 12 rank ( Ω G A ) = 12 rank ( Ω G B )= 1 . These types of calculations become important later in this paper because they allow us to calculate thenumber of nonlocal information qubits in a given set of generators.3.2 EbitsAn ebit is a special case of a nonlocal information qubit where α = β = 1 / √
2. Let | Φ (cid:105) AB denote thestate of an ebit where | Φ (cid:105) AB ≡ | (cid:105) AB + | (cid:105) AB √ . The stabilizer operators of an ebit are Z A Z B and X A X B and, thus, it has error correction capabilityonly. It can detect local errors of the form X A , Z A , X B , and Z B .The binary representation of the stabilizer operators of an ebit are as follows: H ≡ (cid:20) (cid:12)(cid:12)(cid:12)(cid:12) (cid:21) , ≡ (cid:2) H AZ H BZ (cid:12)(cid:12) H AX H BX (cid:3) . Defining H A , H B , Ω H A , and Ω H B similar to the way we did in (2) and (3), it is straightforward to showthat 12 rank ( Ω H A ) = 12 rank ( Ω H B ) = 1 . This result is expected because an ebit contains exactly one ebit of entanglement and the above matrixrank calculation is equivalent to the bipartite entanglement measure of Fattal et al. [22]. This entangle-ment measure, in turn, coincides with the von Neumann entropy of entanglement measure.2.3 Nonlocal Information Qubits versus EbitsThe nonlocal information qubit is a hybrid resource for nonlocal quantum redundancy in a quantumerror-correcting code. It mixes the abilities of an ebit and an information qubit, because it possessesboth error detection capability and information coding ability. That is, it can encode exactly one qubitinto the nonlocal subspace spanned by the states {| (cid:105) AB , | (cid:105) AB } , while at the same time detectinglocal errors of the form X A and X B . In contrast, an ebit is only useful as an error correction resourcebecause it cannot encode arbitrary quantum information.The power of a nonlocal information qubit as used in a quantum error-correcting code lies in-betweenthat of an information qubit and that of an ebit (as discussed above). Thus, there is a qualitative sense in which it is similar to a coherent bit channel [28,54,53], because a coherent bit channel is a resourcewith communication power in-between a noiseless qubit channel and a shared, noiseless ebit. Considerthat the resource of a noiseless information qubit is qualitatively similar to a noiseless qubit channelbecause it arises from the ability to simulate a noiseless qubit channel. That is, there is some means bywhich a source S can distribute an information qubit to Alice with the noiseless qubit channel | x (cid:105) S → | x (cid:105) A , where x ∈ { , } , if she possesses the noiseless information qubit | ψ (cid:105) A = α | (cid:105) A + β | (cid:105) A . Similarly,the resource of a nonlocal information qubit is qualitatively similar to a noiseless coherent bit channelbecause it arises from the ability to simulate a noiseless coherent bit channel, where we define a coherentbit channel as the following isometric map: | x (cid:105) S → | x (cid:105) A | x (cid:105) B , where x ∈ { , } . The input system to the coherent bit channel is a source system S , and the outputsystems are those of Alice and Bob. In particular, the map maintains coherent superpositions, from whichit gains its name as the coherent bit channel. We leave it open as to who possesses the source systembecause it could be Alice (as originally defined in Ref. [28]) or Bob, or some other system. So, we assumethat there is some means by which a noiseless coherent bit channel is simulated if Alice and Bob possessa nonlocal information qubit, i.e., there is some means by which the following map occurs: α | (cid:105) S + β | (cid:105) S → | ϕ (cid:105) AB , liceBob Claire E A E B N B N A | ϕ (cid:31) AB | Φ (cid:31) AB | (cid:31) A | ψ (cid:31) A | ψ (cid:31) B | (cid:31) B D Fig. 1 (Color online) The operation of an [[8 , , ,
1; 1]] two-sender one-receiver quantum error-correcting code. Alice wouldlike to send the information qubit | ψ (cid:105) A , Bob would like to send the information qubit | ψ (cid:105) B , and both Alice and Bob wouldlike to send the nonlocal information qubit | ϕ (cid:105) AB . For quantum redundancy, Alice uses an ancilla | (cid:105) A , Bob uses an ancilla | (cid:105) B , and they both exploit an ebit in the state | Φ (cid:105) AB . They both transmit their encoded state over respective noisyquantum channels N A and N B connecting them to Claire, and she then decodes the three information qubits. where | ϕ (cid:105) AB is defined in (1). Here, we do not concern ourselves with how they happen to come uponnonlocal information qubits, but we merely assume that they have a supply and would like to transmitthem to the receiver Claire.In further analogy of the nonlocal information qubit with the coherent bit channel, an ebit ariseswhen we send a qubit in the state ( | (cid:105) + | (cid:105) ) / √ We now give our model for a bipartite quantum error correction protocol by constructing a bipartitecode from a stabilizer quantum code. We assume that Alice would like to send k A information qubits,Bob would like to send k B information qubits, and both Alice and Bob would like to send k AB nonlocalinformation qubits. An [[ n, k A , k B , k AB ; c AB ]] two-sender one-receiver quantum error-correcting code isone that exploits n total channel uses and c AB ebits shared between Alice and Bob to send the afore-mentioned amounts of information qubits. Figure 1 depicts the operation of an [[8 , , ,
1; 1]] two-senderone-receiver quantum code.Let us consider a set of commuting generators that forms a valid stabilizer code [24]. Suppose thatwe have an [ n, k ] stabilizer code S with generators g , . . . , g n − k . Suppose that we also have the n + k generators of the normalizer N ( S ) of this code. It is possible to determine the logical operators X , . . . , X k , Z , . . . , Z k of this code by performing a symplectic Gram-Schmidt orthogonalization procedure onthe generators of the normalizer N ( S ) [48]—this procedure gives the 2 k logical operators and the n − k stabilizer generators.Now let us assume that Alice possesses the first n A qubits and Bob possesses the last n B where n A + n B = n . We represent the stabilizer generators as follows: g ( A )1 ⊗ g ( B )1 , . . . , g ( A ) n − k ⊗ g ( B ) n − k . The logical operators are as follows: X ( A )1 ⊗ X ( B )1 , . . . , X ( A ) k ⊗ X ( B ) k ,Z ( A )1 ⊗ Z ( B )1 , . . . , Z ( A ) k ⊗ Z ( B ) k , where the A superscript indicates the part of the generator corresponding to qubits that Alice possesses,and the B superscript indicates the part corresponding to qubits that Bob possesses. We can alternativelyrepresent the stabilizer generators with an ( n − k ) × n binary check matrix H where H ≡ (cid:2) H Z (cid:12)(cid:12) H X (cid:3) , (4)5nd the vertical bar divides the matrix into a “Z” part and an “X” part according to the Pauli-to-binary isomorphism (See Ref. [24]). We can further divide the matrix H into matrices corresponding togenerators acting on Alice and Bob’s respective qubits: H ≡ (cid:2) H AZ H BZ (cid:12)(cid:12) H AX H BX (cid:3) , (5)where H AZ and H AX are each ( n − k ) × n A binary matrices and H BZ and H BX are each ( n − k ) × n B binarymatrices. Let H A denote the following matrix: H A ≡ (cid:2) H AZ (cid:12)(cid:12) H AX (cid:3) , (6)and let H B denote the following matrix: H B ≡ (cid:2) H BZ (cid:12)(cid:12) H BX (cid:3) . (7)We can represent the normalizer N ( S ) with an ( n + k ) × n binary matrix G where G ≡ (cid:2) G Z (cid:12)(cid:12) G X (cid:3) . (8)There is a similar subdivided representation of the matrix G : G ≡ (cid:2) G AZ G BZ (cid:12)(cid:12) G AX G BX (cid:3) . (9)Let G A denote the following matrix: G A ≡ (cid:2) G AZ (cid:12)(cid:12) G AX (cid:3) , (10)and let G B denote the following matrix: G B ≡ (cid:2) G BZ (cid:12)(cid:12) G BX (cid:3) . (11)The rowspace of matrix H is in the rowspace of matrix G because the generators in group N ( S ) normalizethe generators in the group S .We can formulate several symplectic product matrices [50,48] that are useful for determining thelocal anticommutativity in the above generators. Let Ω H A be the “Alice” symplectic product matrixcorresponding to Alice’s local matrix H A : Ω H A ≡ H AZ (cid:0) H AX (cid:1) T + H AX (cid:0) H AZ (cid:1) T , where addition is binary. Let Ω H B , Ω G , Ω G A , and Ω G B denote similar symplectic product matricescorresponding to matrices H B , G , G A , and G B .We can manipulate the generators of the stabilizer S into a form more suitable for representationas a bipartite code (this manipulation is similar to that of Theorem 1 in Ref. [22]). We freely abuseterminology by referring to a subgroup by its generating set. We first separate the generators in thestabilizer S into two subgroups with generators of the following forms: S (cid:48) ≡ (cid:110) g ( A ) i ⊗ g ( B ) i (cid:111) ,S B ≡ (cid:110) I ( A ) ⊗ g ( B ) j (cid:111) . (12)It is possible to bring all generators into one of these two subgroups because the local “Alice” generatorsof S B are dependent on the local “Alice” generators of S (cid:48) . We further manipulate the generators todivide into three subgroups: S AB ≡ (cid:110) g ( A ) i ⊗ g ( B ) i (cid:111) ,S A ≡ (cid:110) g ( A ) j ⊗ I ( B ) (cid:111) ,S B ≡ (cid:110) I ( A ) ⊗ g ( B ) p (cid:111) , (13)by using the generators in S B to remove any dependence of the local “Bob” generators of S (cid:48) . We finallybring the generators of S into the following four subgroups by performing the symplectic Gram-Schmidt6rthogonalization procedure (see Ref.’s [50,48]) on the local “Alice” part of the generators in S AB . Thislast step further divides the subgroup S AB into two subgroups S AB E and S AB NLI : S AB E ≡ (cid:40) g ( A ) i ⊗ g ( B ) i g ( A ) i ⊗ g ( B ) i (cid:41) ,S AB NLI ≡ (cid:110) g ( A ) j ⊗ g ( B ) j (cid:111) ,S A ≡ (cid:110) g ( A ) p ⊗ I ( B ) (cid:111) ,S B ≡ (cid:110) I ( A ) ⊗ g ( B ) q (cid:111) . (14)The entanglement subgroup S AB E consists of those generators in S AB which have a locally anticom-muting partner in S AB , where the anticommutativity is with respect to the local “Alice” part of thegenerators. We denote a generator in S AB E by g ( A ) i ⊗ g ( B ) i and its locally anticommuting partner by g ( A ) i ⊗ g ( B ) i . The generators in S AB E therefore correspond to ebits that Alice and Bob share before thequantum communication protocol begins.The nonlocal information subgroup S AB NLI consists of those generators in S AB which have no suchlocally anticommuting partner in S AB . Its locally anticommuting partners are therefore in the normalizer N ( S ). The generators in S AB NLI therefore correspond to nonlocal information qubits that Alice and Bobshare before the quantum communication protocol begins. The local subgroups S A and S B correspondto ancilla qubits for Alice and Bob.The following theorem shows how to produce an [[ n, k A , k B , k AB ; c AB ]] two-sender one-receiver quan-tum code from any [[ n, k ]] stabilizer code, and furthermore, it computes the parameters k A , k B , k AB , and c AB as a function of the stabilizer group S and the normalizer N ( S ) (it actually uses their correspondingbinary representations). Theorem 1
From any [[ n, k ]] stabilizer code, we can produce an [[ n, k A , k B , k AB ; c AB ]] two-sender one-receiver quantum code by choosing Alice to possess n A qubits and choosing Bob to possess n B qubits,where n A + n B = n . The two-sender one-receiver quantum code requires c AB ebits where c AB = 12 rank ( Ω H A ) = 12 rank ( Ω H B ) . It transmits k A information qubits for Alice, k B information qubits for Bob, and k AB nonlocal informationqubits where k AB = rank (cid:0) H A (cid:1) + rank (cid:0) H B (cid:1) + k − n − c AB ,k A = 12 rank ( Ω G A ) − c AB − k AB ,k B = 12 rank ( Ω G B ) − c AB − k AB . We can also compute k AB with the following formula: k AB = 12 ( rank ( Ω G A ) + rank ( Ω G B ) − rank ( Ω G )) − c AB . Proof
The method of proof is similar to that originally posted in Ref. [50] and later exploited in Ref. [48].In this proof, we liberally go back and forth between the binary representation of Pauli generators andthe groups generated by Pauli generators.The number c AB of ebits that the code requires is equal to the number of locally anticommuting pairsin S , with respect to either Alice’s local part or Bob’s local part. Ref. [50] shows that we can calculatethe amount of anticommutativity in any set of generators by calculating the rank of its correspondingsymplectic product matrix and dividing by two. Thus, the number of ebits that the code requires is equalto c AB = 12 rank ( Ω H A ) = 12 rank ( Ω H B ) . We now calculate the number k AB of nonlocal information qubits. The number of generators in S , orequivalently, the number of rows in H , is equal to n − k . The size (cid:12)(cid:12) S B (cid:12)(cid:12) of the local subgroup S B is equal7o n − k reduced by the size | S (cid:48) | of the subgroup S (cid:48) in (12). The size | S (cid:48) | of the subgroup S (cid:48) in (12) isequal to rank (cid:0) H A (cid:1) because it consists of all the locally independent generators. Thus, the size (cid:12)(cid:12) S B (cid:12)(cid:12) is (cid:12)(cid:12) S B (cid:12)(cid:12) = n − k − rank (cid:0) H A (cid:1) . A symmetric argument gives that (cid:12)(cid:12) S A (cid:12)(cid:12) = n − k − rank (cid:0) H B (cid:1) . These results then imply that the size (cid:12)(cid:12) S AB (cid:12)(cid:12) of the subgroup S AB in (13) is (cid:12)(cid:12) S AB (cid:12)(cid:12) = n − k − (cid:12)(cid:12) S A (cid:12)(cid:12) − (cid:12)(cid:12) S B (cid:12)(cid:12) = rank (cid:0) H A (cid:1) + rank (cid:0) H B (cid:1) + k − n. We obtain the number k AB of nonlocal information qubits, or equivalently, the size | S AB NLI | of the nonlocalinformation subgroup S AB NLI , by reducing (cid:12)(cid:12) S AB (cid:12)(cid:12) by the number c AB of ebits: k AB = (cid:12)(cid:12) S AB (cid:12)(cid:12) − c AB = rank (cid:0) H A (cid:1) + rank (cid:0) H B (cid:1) + k − n − c AB . We now calculate the number k A of “Alice” local information qubits. For this task, we considerAlice’s part G A of the full normalizer matrix G . The anticommutativity in the generators correspondingto the rows of the matrix G A is all and only due to ebits, nonlocal information qubits, and Alice’slocal information qubits. The anticommutativity from ebits is due to the local “Alice” generators inthe subgroup S AB E , which are themselves in the rowspace of G A . The anticommutativity from nonlocalinformation qubits is due in part to the local “Alice” generators of S AB NLI and the local part of the matchinglocally anticommuting partners in the local normalizer. Both of these local generators are in the rowspaceof G A . Alice’s local information qubits have logical operators which contribute to the anticommutativityas well. We can calculate the overall number of anticommuting pairs due to ebits, nonlocal informationqubits, and “Alice” local information qubits as rank( Ω G A ) / Ω G A ) = k A + c AB + k AB . (15)Reducing the quantity rank( Ω G A ) / c AB of ebits and the number k AB of nonlocalinformation qubits produces the formula for k A in the statement of the theorem. A symmetric argumentgives that 12 rank ( Ω G B ) = k B + c AB + k AB , (16)and, thus, gives the number k B of local information qubits for Bob.We can also calculate k AB by recalling that rank( Ω G ) / Ω G ) = k AB + k A + k B . Combining the above equation with the equations (15) and (16) and solving for k AB gives the followingformula for k AB : k AB = 12 (rank ( Ω G A ) + rank ( Ω G B ) − rank ( Ω G )) − c AB . Finally, it is possible to produce local encoding circuits for the resulting two-sender one-receiverquantum code by first bringing the stabilizer generators into the form in (14) and applying the algorithmoutlined in the appendix of Ref. [55] to the local parts of the generators (this algorithm, in turn, derivesfrom the Grassl-R¨otteler algorithm for encoding quantum convolutional codes [26]).8 .0.1 Example of a Bipartite Stabilizer Code
We now detail an example of an [[8 , , ,
1; 3]] two-sender one-receiver quantum error-correcting code.Consider the [[8 , , X I Z I Y Z X YI X Z Z Y X Y II I X Y Z Z Y XZ I Z X I Y Y ZZ Z Z Z X Z Z X .
Let us suppose that Alice possesses the first four qubits and Bob possesses the second four qubits.Inspection of Alice’s local generators reveals that they are an independent set of generators, and thesame holds for Bob’s local generators. Thus, the local subgroups S A and S B are empty.We then perform the symplectic Gram-Schmidt orthogonalization procedure on Alice’s local genera-tors and produce the following set of generators: X I Z I Y Z X YI I X Y Z Z Y XI X Z Z Y X Y IZ I Y Z Z X I YI Y Z X Z I Z I .
Notice that the first two generators form a locally anticommuting pair, the second two generators form alocally anticommuting pair, and the last generator does not have an anticommuting partner. Thus, thefirst four generators generate the entanglement subgroup, and the last generator generates the nonlocalsubgroup.Alice and Bob can each then perform local Clifford operations to reduce the above stabilizer to thefollowing trivial one:
Z I I I Z I I IX I I I X I I II Z I I I Z I II X I I I X I II I Z I I I Z I .
In the above stabilizer, we can plainly see that the first four generators correspond to two ebits thatAlice and Bob share, and the last generator corresponds to a nonlocal information qubit (recall fromSection 2.1 that the stabilizer of a nonlocal information qubit is ZZ ). The operators acting on thefourth and eighth qubits are the identity for all stabilizer generators so that Alice can encode one localinformation qubit and Bob can encode one local information qubit. The error-correcting properties ofthe code are equivalent to the error-correcting properties of the original stabilizer code. The bipartite coding structure outlined in this paper is useful in fault-tolerant quantum computation[44,3,23]. This usefulness is due to the following two factors:1. Bipartite codes derived from stabilizer codes maintain their error-correcting properties.2. The encoding circuit consists of ebit preparations and local encoding circuits. For the example code inthis section, the encoding circuit requires only nearest-neighbor interactions and has fewer malignantpairs than the encoding circuit for the baseline Steane code [3].In this section, we represent the Steane code [40] as a bipartite quantum code and show how thisrepresentation gives a simplified, local encoding circuit. As a result, the simplified encoding circuitaffects the “pseudothreshold” [46] for fault-tolerant quantum computation with the Steane code, underthe assumption that quantum memory errors do not occur as frequently as quantum gate errors. Wepresent the results of numerical simulations that demonstrate how the pseudothreshold improves undercertain assumptions. 9 bitsLocal Information Qubit
Fig. 2 (Color online) The encoding operations for the 3EA Steane code. We first prepare the seven-qubit register withthree ebits and one local information qubit (the red qubits belong to the “inside party” and the blue qubits belong tothe “outside party”). The black bars represent “wait gates,” that model potential memory errors that may occur. Theencoding consists of three rounds of local CNOT gates. Note that we place extra “memory decoherence” on the ebits tomodel preparation errors that may occur.
I I I X X X XI X X I I X XX I X I X I XI I I Z Z Z ZI Z Z I I Z ZZ I Z I Z I Z . (17)We can write this code as a bipartite code by employing Theorem 1. In particular, let us give the first,second, and fourth qubits to an “outside” party, and the third, fifth, sixth, and seventh qubits to an“inside” party. Figure 3 makes this nomenclature of “inside” and “outside” more clear. This bipartitecut yields a [[7 , , ,
0; 3]] bipartite quantum code by employing the calculations in Theorem 1. It encodesone local information qubit with the help of three ebits shared between the inside party and the outsideparty. Note that this code is also a [[4 , ,
3; 3]] entanglement-assisted code. We refer to this slight variationof the Steane code as the three-ebit entanglement-assisted Steane code ( ).4.2 Encoding MethodThe advantage of the 3EA Steane code is that it is possible to encode it using only nearest-neighborinteractions on four qubits (if there is a good source of ebits available). This property is desirable fora fault-tolerant encoding circuit because the locality property ensures that errors propagate only tofour qubits during encoding, under the assumption that gate errors occur more frequently than memoryerrors.We now show how to encode the 3EA Steane code with local operations. Suppose that we initialize aseven-qubit quantum register with three ebits and one information qubit. We assume that the particulartechnology implementing the code possesses a good source of ebits, though note that we allow for memoryerrors to occur on both halves of the ebits after they have been prepared. The stabilizer corresponding Note that this distinction between parties is not particularly relevant in fault-tolerant quantum computation, but wemake this distinction in order to appeal to the coding structure outlined before. ,31,3 22(a) (b) Fig. 3
The above figure depicts a particular geometric layout for encoding a set of qubits with the Steane code. The fourinner qubits belong to the “inside” party and the three outer qubits belong to the “outside” party. (a) Three ebits surroundan information qubit before the encoding takes place. (b) The encoding operations are local CNOT gates that interact theinformation qubit and the three local halves of the ebits. The arrows indicate the direction of the CNOT gates, and thenumber adjacent to an arrow indicates the time step at which the operation takes place. to the unencoded state is as follows:
I I I X X I II X X I I I IX I I I I I XI I I Z Z I II Z Z I I I IZ I I I I I Z .
The first round of encoding applies a CNOT gate from the third qubit to the fifth and from the sixthqubit to the seventh. The second round of encoding applies a CNOT gate from the seventh qubit to thethird and from the fifth qubit to the sixth. The CNOT gates in the final round are the same as those inthe first round. The result is that the stabilizer of the encoded state is as given in (17). Figure 2 depictsthe encoding circuit in quantum circuit notation (with a permutation of the qubits for a simplified visualpresentation). Note that that circuit includes memory errors that may occur on the outside party’s shareof the ebits (as explained in the caption of Figure 2). Figure 3 gives an alternative illustration of theencoding circuit that depicts a particular geometric layout of the qubits in the seven-qubit quantumregister.An analysis of this encoding circuit with publicly available fault-tolerant QASM tools [13,12] showsthat it has 9,577 CNOT-CNOT malignant pairs, which are quite a bit fewer than the 13,245 CNOT-CNOT malignant pairs in the AGP Steane encoding circuit [3]. We then expect that the 3EA Steanecode should outperform the baseline Steane code, when quantum memory errors occur less frequentlythan quantum gate errors. The results in the next section confirm this intuition.4.3 Simulation ResultsWe simulated the performance of the 3EA Steane code using publicly available QASM fault-tolerantsimulation software [13,12]. In particular, we evaluated the encoding circuit of the 3EA Steane codein the CNOT extended rectangle (See Figure 11 of Ref. [3]). The CNOT extended rectangle performs“Steane error correction” [45] of two logical qubits (recall that “Steane error correction” is differentfrom the “Steane code”), performs a logical CNOT between the two blocks, and performs Steane errorcorrection again on the two blocks.Figure 4 plots the results of the simulations with accompanying explanations. The result is thatthe 3EA Steane code gives an improvement in performance over the baseline Steane code, under theassumption that quantum memory errors occur less frequently than quantum gate errors.
The bipartite stabilizer formalism represents a new way of thinking about quantum error correction codes.Our main theorem shows how to divy up the qubits in a bipartite quantum code as local information11 a) (b)(c) (d) −4 −1
567 x 10 −4 p p −4 −1
567 x 10 −4 p p −4
56 x 10 −4 p p −4 −1 −4 p p Memory Error Rate - 1x10-4Pseudothreshold - 1.45x10-4Memory Error Rate - 1x10-4Pseudothreshold - 1.74x10-4 Memory Error Rate - 1x10-5Pseudothreshold - 2.34x10-4Memory Error Rate - 1x10-5Pseudothreshold - 1.52x10-4
Fig. 4 (Color online) Various simulation results for the Steane code in the CNOT extended rectangle. Each plot graphsthe probability of failure with encoding versus the probability of failure without encoding. The red line on each plot isthe probability of failure when the encoding is trivial (i.e., there is no encoding). Each blue error bar is the result of 10 simulations, and there are 100 blue error bars on each plot. The black curve going through the error bars is a quadraticfit to the data points. The dashed line on each plot indicates the location of the pseudothreshold (the point at which thered curve intersects the black fitted curve). The result is that the 3EA Steane code gives an improvement over the baselineSteane code when memory errors occur less frequently than gate errors. (a) Baseline Steane code with memory error rate1 × − . An estimate of the pseudothreshold is 1 . × − . (b) Baseline Steane code with memory error rate 1 × − . Anestimate of the pseudothreshold is 1 . × − . (c) 3EA Steane code with memory error rate 1 × − . An estimate of thepseudothreshold is 1 . × − . (d) 3EA Steane code with memory error rate 1 × − . An estimate of the pseudothresholdis 2 . × − . qubits, nonlocal information qubits, ancilla qubits, and ebits. Our original purpose was to show how thebipartite stabilizer formalism is useful for quantum communication, but it turns out to have use in fault-tolerant quantum computation as well. In particular, the 3EA Steane code improves the pseudothresholdfor fault-tolerant quantum computation, under the assumption that there is a good source of ebitsand quantum memory errors occur less frequently than quantum gate errors. This broad applicabilityreinforces the strong links between techniques for quantum communication and techniques for quantumcomputation.We now list several open problems of interest for extensions in quantum communication topics. – It would be interesting to develop quantum Shannon-theoretic protocols that include nonlocal infor-mation qubits. The state merging protocol may be useful here [29,30]. – The codes in this paper are relevant for a multiple-access channel, but it could be interesting todetermine if there are useful codes for a broadcast channel. Yard et al . have already explored quantum12hannon theoretic protocols for the quantum broadcast channel [58]. Perhaps techniques from thatwork will give insight into the design of quantum broadcast channel codes. – It might also be interesting to explore bipartite convolutional codes as an extension of the work inRef’s. [55,49,51] and the work in this paper. – There might be a way to develop multiparty codes that exploit a common secret key between multipleparties because of the connection between quantum privacy and quantum coherence [14,19,18]. – Finally, we are currently considering the extension of the ideas in this paper to the tripartite settingwhere the senders share quantum resources with the receiver (this extension would be similar to theway that Ref. [5] extends Ref. [22]).It should also be interesting to explore further improvements that might arise in fault-tolerant sim-ulations of the 3EA Steane when taking into account the nearest-neighbor interactions in its encodingcircuit, similar to the way that Ref. [38] considered the impact of nearest-neighbor interactions for quan-tum Fourier transform circuits.The authors thank Todd Brun, Min-Hsiu Hsieh, and Ognyan Oreshkov for extensive feedback on themanuscript. MMW acknowledges support from the MDEIE (Qu´ebec) PSR-SIIRI international collabo-ration grant.
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