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AIP Advances / Volume 1 / Issue 4 / REGULAR ARTICLES

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AIP Advances 1, 042172 (2011); http://dx.doi.org/10.1063/1.3672009 (29 pages)

Categorical Tensor Network States

Jacob D. Biamonte1,2,3, Stephen R. Clark2,4, and Dieter Jaksch2,4,5

1Oxford University Computing Laboratory, Parks Road Oxford, OX1 3QD, United Kingdom
2Centre for Quantum Technologies, National University of Singapore, 3 Science Drive 2, Singapore 117543, Singapore
3ISI Foundation, Torino Italy
4Keble College, Parks Road, University of Oxford, Oxford OX1 3PG, United Kingdom
5Clarendon Laboratory, Department of Physics, University of Oxford, Oxford OX1 3PU, United Kingdom

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(Received 25 June 2011; accepted 18 October 2011; published online 12 December 2011)

We examine the use of string diagrams and the mathematics of category theory in the description of quantum states by tensor networks. This approach lead to a unification of several ideas, as well as several results and methods that have not previously appeared in either side of the literature. Our approach enabled the development of a tensor network framework allowing a solution to the quantum decomposition problem which has several appealing features. Specifically, given an n-body quantum state |ψ〉, we present a new and general method to factor |ψ〉 into a tensor network of clearly defined building blocks. We use the solution to expose a previously unknown and large class of quantum states which we prove can be sampled efficiently and exactly. This general framework of categorical tensor network states, where a combination of generic and algebraically defined tensors appear, enhances the theory of tensor network states.

© 2011 Author(s). This article is distributed under a Creative Commons Attribution 3.0 Unported License.

Article Outline

  1. INTRODUCTION
  2. RESULTS OVERVIEW
    1. Tensor network representations of quantum states
    2. Network components fully defined by diagrammatic laws
    3. Boolean and multi-valued tensor network states
    4. Putting it all together: connecting the dots
  3. CONSTITUENT NETWORK COMPONENTS: A TENSOR TOOL BOX
    1. COPY -tensors: the “diagonal”
    2. XOR -tensors: the “addition”
    3. Generating the affine class of networks
    4. Quantum AND -state tensors: Boolean universality
      1. Summary of the XOR -algebra on tensors
    5. co- COPY : the co-diagonal
    6. The remaining Boolean tensors: NAND -states etc.
    7. Summarizing: network composition of quantum logic tensors
  4. INTERACTION OF THE NETWORK COMPONENTS
    1. Associativity, distributivity and commutativity
    2. Bialgebras on tensors
      1. Hopf algebras on tensors
    3. Bending wires: compact structures
  5. EXAMPLES OF CATEGORICAL TENSOR NETWORK STATES
    1. Constructing Boolean states
    2. Describing states with complex coefficients
  6. PROOF OF THE MAIN THEOREMS
  7. OUTLOOK AND CONCLUDING REMARKS

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KEYWORDS and PACS

Keywords

category theory, quantum theory

PACS

ARTICLE DATA

Digital Object Identifier
http://dx.doi.org/10.1063/1.3672009

PUBLICATION DATA

ISSN

2158-3226 (online)

Publisher:

American Institute of Physics is a Member of CrossRef is a Member of CrossRef American Institute of Physics

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Figures (24) Tables (4)

Figures (click on thumbnails to view enlargements)

FIG.1
(a) A generic quantum state |ψ〉 for n degrees of freedom represented as a tensor with n open legs. (b) A comb-like MPS tensor network for a 1D chain system.51 , 52 (b) A grid-like PEPS tensor network for a 2D lattice system.39 , 53 (d) A TTN for a 1D chain system where only the bottom layer of tensors possess open physical legs.55 , 56 (e) A TTN for a 2D lattice system. (f) A hierarchically structured MERA network for a 1D chain system possessing unitaries (rank-4 tensors) and isometries (rank-3 tensors).25 , 54 This tensor network can also be generalized to a 2D lattice (not shown).

FIG.1 Download High Resolution Image (.zip file) | Export Figure to PowerPoint

FIG.2
(a) One of the simplest tensors, called the diagonal in category theory, the COPY-gate or the COPY-dot in circuits, copies computational basis states |x〉 where x = 0, 1 for qubits and x = 0, 1, …, d − 1 for qudits. The tensor subsequently breaks up into disconnected states. (b) A generic PEPS in which we expose a single generic rank-5 tensor. This tensor network can neither be contracted nor sampled exactly and efficiently. However, if the tensor has internal structure exploiting the COPY-dot then efficient sampling becomes possible. (c) The tensor breaks up into a vertical and a horizontal rank-3 tensor joined by the COPY-dot. Upon sampling computational basis states the resulting contraction reduces to many isolated MPS, each of which are exactly contractible, for each row and column of the lattice. This type of state is known as a string-bond state and can be readily generalized.57 (d) An even simpler case is to break the tensor up into four rank-2 tensors joined by a COPY-dot forming a co-called correlator-product state.59 (e) Finally, outside the PEPS class, there are entangled plaquette states58 which join up overlapping tensors (in this case rank-4 ones describing a 2 × 2 plaquette) for each plaquette. Efficient sampling is again possible due to the COPY-dot.

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FIG.3
Read top to bottom. A presentation of the linear fragment of the Boolean calculus. The plus (⊕) dots are XOR and the black (•) dots represent COPY. The details of (a)-(g) will be given in Sections 3 , 4. For instance, (d) represents the bialgebra law and (g) the Hopf-law (in the case of qubits x ⊕ x = 0, in higher dimensions the units 〈+| becomes 〈0| + 〈1| + ⋅⋅⋅ + 〈d − 1|).

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FIG.4
Diagrams read top to bottom. A presentation of the Boolean-calculus with Figure 3. The details of (a)-(g) will be given in Sections 3 , 4. For instance, (h) represents distributivity of AND (∧) over XOR (⊕), and (d) shows that x ∧ x = x.

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FIG.5
Example of the Boolean quantum AND-state or tensor. In (a) the network is run backwards (post-selected) to 〈1| resulting in the product state |11〉. In (b) the tensor is post-selected to 〈0| resulting in the entangled state |00〉 + |01〉 + |10〉.

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FIG.6
A general multi-valued qudit state based on a d-switching function f can either be formed as (a) by inputting a logical-one at the output of the circuit as described by Equation ( 1 ) or (b) by bending the output of the circuit around to form an input as in Equation ( 2 ). For multi-valued qudit states, the boolean function f:math2n→math2 becomes a multi-valued qudit function f:mathdn→mathd. The network (a) is then post-selected to α0|0〉 + α1|1〉 + ⋅⋅⋅αd−1|d − 1〉 where ∀i, αi = 0/1.

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FIG.7
Salient diagrammatic properties of the COPY-dot. (a) Full-symmetry. (b) Copy points, e.g. |x〉 ↦ |xx〉 for x = 0, 1 for qubits and x = 0, 1, …, d − 1 for d dimensional qudits. (c) The unit — in this case the unit corresponds to deletion, or a map to the terminal object which is given as 〈+|math〈0|+〈1| for qubits and 〈+|math〈0|+〈1|+⋯+〈d−1| for d dimensional qudits (the bi-direction of time is explained later by considering co-diagonals in Section 3E). (d) Co-interaction with the unit creates a Bell state ∑i = 0d−1|ii〉. This and the corresponding dual under the dagger form the compact structures of the †-category of quantum theory.

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FIG.8
Salient diagrammatic properties of the AND-tensor. (a) Input-symmetry. (b) Existence of a zero or fixed-point. (c) The unit |1〉. (d) Co-interaction with the unit creates a product-state. Note that the gate forms a valid quantum operation when run backwards as in (d).

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FIG.9
Illustrates the use of units to prepare the AND-state. Using this state together with single qubit NOT-gates, one can construct any Boolean qubit state as well as any of the states appearing in Table 4. We note that the box around the Toffoli gate (left) is meant to illustrate a difference between our notation and that of quantum circuits.

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FIG.10
Hadamard built from the AND-state together with |−〉mathmath(|0〉−|1〉). We note that quantum computational universality is already possible by considering simple Hadamard states (e.g. |ψH〉 = |00〉+|01〉+|10〉−|11〉), COPY- and AND-states, which follows from the proof that Hadamard and Toffoli are universal for quantum circuits.67

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FIG.11
Diagrammatic equations satisfied by a fixed point pair (see Definition 9).

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FIG.12
AND and OR tensors form a fixed point pair. The unit for AND (|1〉 see a) is the zero for OR (c) and vise versa: the unit of OR (|0〉 see a) is the zero for AND (b).

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FIG.13
Node equivalence or spider law.68 Connected black-dots (•) as well as connected plus-dots (⊕) can be merged and also split apart at will. The intuition for digital or qudit circuits follows by connecting a state |ϕ〉 to one of the legs and iterating over a complete basis |0〉, |1〉, …, |d − 1〉.

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FIG.14
The GHZ-state tensor is simply a rank-n COPY-dot. Node equivalence implies that this tensor can be deformed into any network geometry including a MPS comb-like structure (right).

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FIG.15
Reduced density operator. Left (a) reduced density operator ρGHZ′ found from applying the node equivalence law to a n-qubit GHZ-state. Right (b) the expectation value of observable O1 ⊗ O2 found from connecting the observable and connecting the open legs (i.e. taking the trace).

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FIG.16
Bialgebra axioms68 (scalars are omitted). (a) unit laws (these are of course left and right units); (b) associativity; (c) bialgebra; (d,e) co-COPY points.

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FIG.17
Cup identities. (a) Symmetry. (b) Conjugate state. (c) Teleportation61 or the snake equation. (d) Sliding an operator around a cup transposes it.

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FIG.18
Diagrammatic adjoints. Cups and caps allow us to take the transpose of a linear map. Note that care must be taken, as flipping a ket |ψ〉 to a bra 〈ψ| is conjugate transpose, and bending a wire is simply transposition, so the conjugate must be taken: e.g. acting on |ψ〉 with a cap given as ∑i〈ii| results in 〈math|.

FIG.18 Download High Resolution Image (.zip file) | Export Figure to PowerPoint

FIG.19
Left (a) the circuit realization (internal to the triangle) of the function fW of e.g. ( 23 ) which outputs logical-one given input |x1x2x3〉 = |001〉, |010〉 and |100〉 and logical-zero otherwise. Right (b) reversing time and setting the output to |1〉 (e.g. post-selection) gives a network representing the W-state. The naïve realization of fW is given in Figure 21 with an optimized co-algebraic construction shown in Figure 21.

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FIG.20
Naïve CTNS realization of the familiar W-state |001〉 + |010〉 + |100〉. A standard (temporal) acyclic classical circuit decomposition in terms of the XOR-algebra realizes the function fW of three bits. This function is given a representation on tensors. As illustrated, the networks input is post selected to |1〉 to realize the desired W-state.

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FIG.21
W-class states in the categorical tensor network state formalism. (a) is the standard W-state. (b) is found from applying De Morgan's law (see Section 3D) to (a) and rearranging after inserting inverters on the output legs. Notice the atemporal nature of the circuits, as one gate is used forwards, and the other backwards.

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FIG.22
W-state (n-party) in the categorical tensor network state formalism. The comb-like feature of efficient network contraction remains, with the internal structure of the network components exposed in terms of well understood algebraic structures.

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FIG.23
Categorical tensor network representing state |ψ〉 = |01〉 + |10〉 + αk|11〉, as explained in Example 22.

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FIG.24
The CTNS for a state |ψ〉 resulting from our exhaustive construction procedure.

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Tables

Table I. Summary of the COPY-gate from Section 3A.

View Table
Table II. Summary of the XOR-gate from Section 3B.

View Table
Table III. Summary of the AND-gate from Section 3D.

View Table
Table IV. The bit pattern of these quantum states represents a Boolean function (given by the subscript) such that the right most bit is the Boolean functions output, and the two left bits are the functions inputs, and the non-linear Boolean functions are on the left side of the table and the linear functions on the right. Consider the state |ψAND〉, and Boolean variables x1 and x2, then the superposition |ψAND〉 encodes the function |x1, x2, x1 ∧ x2〉 in each term in the superposition, and |ψAND〉 = ∑x1,x2∈{0,1}|x1,x2,x1∧x2〉. As outlined in the text, cup/cap induced-duality allows us (for instance) to express this state as the operator |0〉〈00| + |0〉〈01| + |0〉〈01| + |1〉〈11|: : |x1, x2〉 ↦ |x1 ∧ x2〉 which projects qubit states to the AND of their bit value.

View Table




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