A Graphical Calculus for Quantum Computing with Multiple Qudits using Generalized Clifford Algebras
Abstract
In this work, we develop a graphical calculus for multi-qudit computations with generalized Clifford algebras, using the algebraic framework developed in [1]. We build our graphical calculus out of a fixed set of graphical primitives defined by algebraic expressions constructed out of elements of a given generalized Clifford algebra, a graphical primitive corresponding to the ground state, and also graphical primitives corresponding to projections onto the ground state of each qudit. We establish many properties of the graphical calculus using purely algebraic methods, including a novel algebraic proof of a Yang-Baxter equation and a construction of a corresponding braid group representation. Our algebraic proof, which applies to arbitrary qudit dimension, also enables a resolution of an open problem in [2] on the construction of self-dual braid group representations for even qudit dimension. We also derive several new identities for the braid elements, which are key to our proofs. In terms of physics, we connect these braid identities to physics by showing the presence of a conserved charge. Furthermore, we demonstrate that in many cases, the verification of involved vector identities can be reduced to the combinatorial application of two basic vector identities. We show how to explicitly compute various vector states in an efficient manner using algebraic methods. Additionally, in terms of quantum computation, we demonstrate that it is feasible to envision implementing the braid operators for quantum computation, by showing that they are 2-local operators. In fact, these braid elements are almost Clifford gates, for they normalize the generalized Pauli group up to an extra factor , which is an appropriate square root of a primitive root of unity.
2020 Mathematics Subject Classification : 81P68, 81R05.
Keywords— Generalized Clifford algebras, quantum computation, Yang-Baxter equation, and braid group.
1 Introduction
The following physics questions motivate this article: Can we learn new things about quantum entanglement by studying a graphical calculus for the generalized Clifford algebras111The earliest paper introducing generalized Clifford algebras appears to be [3] in 1952. Other early work included [4] in 1964, [5] in 1966, and [6] in 1967.? In this setting, braiding operators defined using the generalized Clifford algebra are unitary operations that entangle neighboring qudits (multi-dimensional vector spaces). Thus, when we apply a sequence of braiding operators to the ground (or vacuum) state, we expect different kinds of entangled states to result, depending on the sequence and on the braidings in the sequence. Is there an easy way to classify the resulting kinds of entanglement using the graphical calculus? How does the classification depend on the number of qudits involved?
To set the stage for a treatment of these questions in a systematic manner, a algebraic framework was presented in [1]. While the algebraic framework is in it of itself sufficient for doing calculations and proving identities of various sorts, it turns out to be convenient to consider diagrammatic representations in order to obtain intuition about what kind of algebraic identities might be true. In contrast to the work of [7], this article will develop the graphical calculus along completely algebraic lines. A new result achieved in this article is an algebraic proof that a particular braid operator satisfies the Yang-Baxter equation, valid over all , which resolves an open question of Cobanera and Ortiz [2] about unitary self-dual braid group solutions for even.
To enable users of the graphical calculus we present to proceed in an entirely algebraic and rigorous way, the following flowchart is presented:
-
1.
Write down an algebraic expression.
-
2.
Convert it to one of the prescribed graphical forms.
-
3.
Guess what graphical identities might be true for the graphical expression.
-
4.
Write down conjectural algebraic identities corresponding to the conjectured graphical identities.
-
5.
Prove the conjectured identities algebraically using explicit calculation with the algebraic framework for the generalized Clifford algebras, or using already proven algebraic identities.
-
6.
Repeat.
It is quite remarkable how far one can get with this approach, once the initial difficulties of getting algebraic identities is overcome. In particular, we show that the algebraic framework, coupled with some new technical innovations of ours, enables us to show algebraically for the first time why one can treat the braiding operator as a braid in the conventional sense (namely, it satisfies a Yang-Baxter equation222One important conceptual and technical point is that the Yang-Baxter equation [8], or rather, a braiding in the tensor categorical sense[9], appears to primarily refer to a morphism from to , where is an algebra, which embeds in . The equation we will prove will have structural similarity to the Yang-Baxter equation, but to truly show that the equation is in fact a Yang-Baxter equation, it is necessary to show that the braid is a 2-local operator. This fact will be proven later in the section on applications. The reason for this subtlety is that generalized Clifford algebras have an additional time-ordering [7] when one wants to “tensor” elements together, and hence there is no global tensor product for the algebra. This additional structure could be useful in its own right.).
For logical consistency, the reader should consider the graphical calculus as simply a transcription of the algebraic framework into a combination of a few basic building blocks, which aids in intuition. While it may be tempting to imagine that the diagrams mean something, the reader will do well to remember that all our proofs are purely algebraic, and the diagrams are just (very helpful) visual aids.
In terms of the graphical representation, the diagrams allowed are a much smaller subset than as those of [7], in order to ensure unambiguous identification of a graphical diagram (via vertical decomposition) with an algebraic expression. In line with the requisite of unambiguity of graphical-to-algebraic correspondence, no independent interpretation is made of the subcomponents of the diagrams. The latter constraint imposed by our work makes its necessary to specify in advance all the possible configurations one may encounter in a full diagram, and the corresponding algebraic expressions. This specification is accomplished using the tool of diagrammatic composition, from the theory of Temperley-Lieb algebras [10], applied to a particular (small) set of graphical primitives which are specified in their completeness.
From a physical perspective, while it has been previously thought [11] that the graphical representation of generalized Clifford algebras is akin to Feynman diagrams, in fact the particular graphical representation considered in this article is more accurately a description of causal diagrams, which arise in the old-fashioned perturbation theory approach to quantum field theory. Thus, the diagrams are more in the spirit of Schwinger’s approach to quantum field theory than Feynman’s, as causality was at the heart of Julian Schwinger’s approach to quantum electrodynamics [12]. On a technical level, whereas the Feynman diagrams of Richard Feynman emphasize propagators in momentum space, Schwinger’s approach emphasized Green’s functions, which are correlation functions in position space.
This correspondence of the graphical representation with a causal description is ensured by the faithful transcription of diagrams into algebraic expressions. In other words, the identification of the time (vertical) axis with the order of operator composition from right to left has been elevated to the role of a physical constraint on the graphical representation. In this sense, the graphical identities that are proven in this article for vectors can be interpreted as showing that certain different unitary processes, when acting on a particular initial state, yield the same final state.
Overall, the results of this article may be summarized as the following: A graphical calculus is presented for multi-qudit computations with generalized Clifford algebras, using the algebraic framework developed in [1]. The graphical calculus is built out of a fixed set of graphical primitives defined by algebraic expressions constructed out of elements of a given generalized Clifford algebra, a graphical primitive corresponding to the ground state, and also graphical primitives corresponding to projections onto the ground state of each qudit. Many graphical properties of the graphical calculus are proven using purely algebraic methods (as well as extended to algebraic identities which are not captured by the graphical representation), including a novel algebraic proof of a Yang-Baxter equation and a construction of a corresponding braid group representation. Our algebraic proof, which applies to arbitrary qudit dimension, also enables a resolution of an open problem in [2] on the construction of self-dual braid group representations for even qudit dimension. Several new identities are derived for the braid elements, including an important relation for bringing a charge over a braid, which are key to the proofs. In terms of physics, these braid identities reflect the presence of a conserved charge. Furthermore, we demonstrate that in many cases, the verification of involved vector identities can be reduced to the combinatorial application of two basic vector identities. We show how to explicitly compute various vector states in an efficient manner using algebraic methods. Additionally, in terms of quantum computation, we demonstrate that it is feasible to envision implementing the braid operators for quantum computation, by showing that they are 2-local operators. In fact, these braid elements are almost Clifford gates, for they normalize the generalized Pauli group up to an extra factor , which is an appropriate square root of a primitive root of unity.
2 The Graphical Calculus
2.1 Building Blocks
The philosophy followed in the graphical calculus we present is that the diagrams drawn are indivisible. No a priori meaning is assigned to the subcomponents of the diagrams, i.e. a single strand, or a single cap, or a single cup. The philosophy adopted is that the algebraic framework of [1] ought to be robust enough that one can derive a posteriori a large number of algebraic relations, and therefore by proving more and more relations, the initially content-free diagrams acquire new, emergent properties. On a technical level, this approach leads to a more basic construction of a graphical calculus which is directly built out of the elements of the generalized Clifford algebra, which is justified by the axiomatic framework.
In devising the graphical representation, we need to consider at the outset what kind of diagrams should be allowed. From the perspective of mathematical rigor, if one proceeds on entirely algebraic grounds, and it is decided to base the manipulation of graphical diagrams on corresponding algebraic identities, it becomes necessary that each graphical diagram have a unique algebraic expression. Note that the word “expression” is used, as opposed to “value.” Two expressions may evaluate to the same algebraic element in the generalized Clifford algebra. Likewise, two graphical diagrams may be different in the sense that they correspond to different algebraic expressions, but equal in the sense that the expressions they correspond to can be shown to be algebraically equal (under the relations of the generalized Clifford algebra and the two axioms).
To be mathematically precise, one has to specify in what sense one means “uniqueness.” In this article, by uniqueness of the algebraic expression corresponding to a diagram, it is meant that the formal algebraic expression (forgetting all properties of the generalized Clifford algebra, except associativity, the property that for any elements of the algebra) obtained from the diagram is invariant under vertical decomposition of the diagram, up to associativity. Thus, the graphical primitives are carefully chosen to guarantee uniqueness of an operator correspondence beyond diagrams and equations, a correspondence which is compatible with the vertical decomposition of diagrams. Adhering to this dictum results in a set of allowed diagrams that is much smaller than that of [7].
In previous work [1], two axioms were presented as a way to abstract certain high-level properties of the generalized Clifford algebras. It was shown that these 2 axioms are satisfied by an explicit construction. These axioms will now be converted into graphical form. As before, let us fix a positive integer greater than 1, a positive integer at least 1, and consider the generalized Clifford algebra generated by , , , , subject to if , and for all . Here, is a primitive th root of unity. When , one recovers the Clifford algebra with generators. For our purposes, we will also need to define satisfying and according to the following lemma.
Lemma 2.1.
Let . If is odd, is the only square root of satisfying . If is even, setting to be either square root of will satisfy .
Let us first define a series of graphical primitives. These graphical primitives are the only allowed graphical elements in our graphical representation. Any diagram encoded using this set of graphical primitives must be specified by a sequence of graphical primitives. One may think of each diagram as a hieroglyph in an alphabet of hieroglyphs, and the sequence of hieroglyph as running from top to bottom. (This corresponds to the composition of operators, in which, in terms of the corresponding algebraic objects, the corresponding algebraic expression are given by a sequence of operations running from right to left.)
Fix . The following graphical primitives are defined in terms of the distinguished ground state (satisfying the two axioms) via:
Definition 2.2.
| (2.1) |
| (2.2) |
Definition 2.3.
| (2.3) |
| (2.4) |
Here we mean for the label to be placed immediately left of the -th strand, and the label to be placed immediately left of the -th strand. There are total strands in each diagram.
We also define for completion that
| (2.5) |
Note that the identity primitive composed with itself “is” itself, graphically, which is consistent with its definition as being equal to 1. Similarly, the identity primitive composed (in either order) with the primitives for the powers of the generators again yields those same primitives. In this sense, the diagrammatic definitions are well-behaved.
Definition 2.4.
| (2.6) |
Here we mean for the “cup-cap” combination to be replacing the and th strands.333In this respect, in our graphical calculus, we do not allow for the cup-cap combination which is prescribed in [7], i.e. we don’t allow not-in-place placement, i.e. on the and th strands, which loosely speaking, straddles different qudits. There are strands in total.
Definition 2.5.
We also define a graphical primitive, which we call the positive braid on strands and , for :
| (2.7) |
| (2.8) |
| (2.9) |
| (2.10) |
which defines different braid operators.
We also define graphical primitives for the corresponding negative braids:
| (2.11) |
| (2.12) |
| (2.13) |
| (2.14) |
The algebraic definition of these braid elements is given by
| (2.15) |
and
| (2.16) |
for in . Here,
| (2.17) |
Note that this is a general definition of the braid element, which goes beyond the diagrams above, since we allow for , which includes the local (nearest-neighbor) braid operators as a special case.
Remark 2.6.
Thus, in terms of terminology, we will refer to the positive braids as just braids, and the negative braids as adjoint braids.
2.2 Graphical Representation of the Axioms
Let us recall the axioms of [1]:
Axiom 1: Let be a complex vector space upon which the generalized Clifford algebra is realized as unitary by matrix operators. Assume that there exists a state (which we call the ground state) which is a tensor of states , , that satisfies the following algebraic identity:
for all , where is a square root of such that .
In addition, for each qudit, the projector onto the th qudit’s ground state is assumed to satisfy
Axiom 2: Scalar product: The set is an orthonormal basis for .
These axioms are now shown to give rise to basic graphical identities. The algebraic identities
for ,
for all , as well as
tell us that
| (2.19) |
i.e. when the primitive for precedes that for , swapping the order of primitives yields a factor of , for , and also that
| (2.20) |
and
| (2.21) |
Furthermore, the vector identity
yields the diagrammatic “identity”
| (2.22) |
An additional identity which is useful [7] is the following:
Lemma 2.7.
| (2.23) |
for , , integers.
Proof.
By double induction on and . ∎
Another identity, due to [7], is
Lemma 2.8.
| (2.24) |
for , an integer.
Proof.
By induction. ∎
3 Algebraic Identities from Algebraic Methods
Our aim in this section is to obtain a large swath of identities, which are related to the graphical representation we have presented, but for which we provide purely algebraic proofs. At the heart of the results of this section are a new “charge-braid” identity that answers an open question due to Jaffe, namely, how to bring the charge “over” the braid when (this terminology will make more sense when we introduce the notion of a conserved charge). This seemingly innocuous result is used to great effect, by using the structural property that the generalized Clifford algebra generated by has trivial center. In particular, we provide an algebraic proof, using the proof strategy based on this structural characterization, that the braid elements satisfy many Yang-Baxter equations. Furthermore, we construct a general solution to the braid group relations, which enables us to resolve an open question of [2] for the case where is even.
3.1 Structural Properties of the Generalized Clifford Algebras
Proposition 3.1.
The set is a basis for the generalized Clifford algebra .
Proof.
Any element of the generalized Clifford algebra is a finite sum of elements of the form for , a positive integer, in the index set , and for . By repeatedly applying the relations and for to swap the order of multiplication, we can put each term in the sum into normal form, by which we mean that the term is of the form , for . Thus, we obtain that every element of the generalized Clifford algebra is prescribed by a sum given by
Now we want to show that in the algebra if and only if for all indices, i.e. the set is a basis. The if direction is obviously true. For the only if direction, suppose . Then multiplying by any product of generators also yields zero. It is clear that we can multiply on the left by the product so that the constant term of is . Thus, without loss of generality, it suffices to show that if , then its constant term must vanish. Then the rest of the coefficients all vanish by applying the same result to
for each index tuple.
To show that the constant term must vanish, we use an operator method. Consider the set of operators , and let and define iteratively. Then the operator acting on a term yields
| (3.25) |
where if , and otherwise. Acting on by the commuting operators (which all have a diagonal action on ) thus projects down to
| (3.26) |
We first claim that the only terms that survive are those for which for . This can be seen since
| (3.27) |
for all implies that
| (3.28) |
for all , and so
| (3.29) |
as desired.
As a result, we further obtain that
since
Finally, using for we obtain that for all . Hence the constant term is the only term left, and must equal since .
∎
Proposition 3.2 (Golden Rule).
The generalized Clifford algebra has trivial center, i.e. the only elements that commute with all elements of the generalized Clifford algebra are .
Proof.
Every element of the generalized Clifford algebra is prescribed by a sum given by
Using the basis property (Proposition 3.1), it becomes simple to show that the algebra has trivial center. Note that the basis property implies uniqueness of the sum decomposition. Let lie in the center of the algebra, and . Then there is an index label such that . Note that implies that by comparing the coefficient of . Thus, . Similarly, implies that and so
| (3.30) |
for from to , where if and if and if , yielding equations in unknowns. Equivalently,
| (3.31) |
for all . Since in Proposition 3.1, it was shown that this set of equations is uniquely solved by , it follows that is a multiple of the identity . ∎
3.2 An “Intertwining” Approach for New Identities for the Generalized Clifford Algebra
3.2.1 A Systematic Procedure
The golden rule of Proposition 3.2 allows us to give a systematic procedure for proving identities in the algebra. The basis of the procedure is the following proposition:
Proposition 3.3.
Let , lie in the generalized Clifford algebra, and suppose is invertible. Further assume that the constant terms of and are nonzero. Then if and only if lies in the center of the generalized Clifford algebra, and the constant term in agrees with the constant term in .
Proof.
Clearly, the only if direction is true since implies . For the if direction, if lies in the center, by the golden rule, , i.e. . In the proof of proposition 3.2, we showed that this implies that all terms of and agree, in particular the constant terms. By hypothesis, the constant terms of and agree and are nonzero, so . ∎
We now provide a concrete way to show that an element lies in the center of the generalized Clifford algebra.
Proposition 3.4.
An element lies in the center of the generalized Clifford algebra if and only if it commutes with for each .
Proof.
The only if direction is clearly true.
For the if direction, any element in the algebra has a unique decomposition as
By iterative commutation, using the commutation property of with , one can show that . Multiplying by the constant prefactor and summing over the indices, one obtains that , as desired, for arbitrary in the algebra. ∎
3.2.2 Intertwining Identities
By intertwining identities, we mean identities of the form . In this section, we prove some new intertwining identities, using the systematic procedure we presented in the previous subsection.
In particular, we have discovered the following new intertwining identity for the braid . We first give a direct proof, and then give an alternate proof which involves some intermediate intertwining identities, which may have more general applications.This identity significantly generalizes a result of [7], which is the special case for .
Proposition 3.5.
| (3.32) |
for .
Proof.
Since , it suffices to show that
By shifting the index of summation from to in the LHS, the LHS becomes
| (3.35) |
which is just the RHS. ∎
In terms of the graphical calculus, we economically write down the following diagrammatic identity, which is specific to and the generalized Clifford algebra with only 2 generators , :
| (3.36) |
It is convenient to also write down the corresponding identity for the adjoint braid:
Corollary 3.6.
| (3.37) |
for , and , integers.
Proof.
The adjoint of the identity in 3.5 is , which becomes upon commutation. Now we let , , so
| (3.38) |
which gives the desired result. ∎
The corresponding diagrammatic identity for the adjoint braid arising from Corollary 3.6 for the generalized Clifford algebra with two generators , is
| (3.39) |
We now pursue an alternate route to proving Equation 3.5, which illuminates complementary aspects. We start with an intertwining identity which is a commutation relation:
Lemma 3.7.
| (3.40) |
for .
We also note that the following commutation relation holds as well:
Lemma 3.8.
| (3.41) |
for and satisfies or .
Proof.
If , commuting past (in front of) in the LHS yields ; commuting it past then yields an additional factor . So we obtain the RHS. A similar proof applies for the case . ∎
Now comes the exciting part. Since the braid is a sum of elements of the form , it follows that
Lemma 3.9.
| (3.42) |
for .
Proof.
By linear extension of Lemma 3.7. ∎
Now we use a simple result due to Jaffe and Liu [7]:
Lemma 3.10.
| (3.43) |
for .
Proof.
It suffices to show that
| (3.44) |
Collecting terms, it is equivalent to show that
| (3.45) |
It is clear that the two are equal since the RHS is just the LHS with shifted to . ∎
3.2.3 The Notion of Charge Conservation
We now interpret the previous section’s intertwining identities in terms of physics. In particular, it is observed that the new charge-braid identity in Proposition 3.5 is a consequence of a particular property of neutral pairings of and . First, we define a charge operator :
Definition 3.11.
Define by linear extension of its action on the basis:
| (3.47) |
for all integer indices . We call the charge of the basis element, following [13], which is well-defined modulo . This terminology of an element’s charge is also applicable for linear combinations of basis elements with the same charge.
Then, lemma 3.7 tells us that eigenstates of of eigenvalue 1 which lie in the subalgebra generated by , commute. We call eigenstates of with eigenvalue 1 neutral.
Graphically, we can describe this commutation relation 3.7 for the algebra generated by and as
| (3.48) |
and there are analogous diagrams (with additional strands in between, and to the left and right) for the generalized Clifford algebras with more generators.
We now observe that the lemma 3.9 can be reinterpreted in terms of respecting charge conservation, i.e. bringing an element of definite charge across the braid will conserve the charge, which is in this case just . Thus, we say that the relation 3.9 provides a physical constraint on the action of the braid. In fact, this physical constraint provides a compelling explanation for why the master intertwining relation 3.5 holds; the latter is essentially forced by the constraint and the additional relation .
3.3 Applications of the Golden Rule
Using the prior sections on the golden rule and various intertwining identities, we can now prove some identities involving the braid in a relatively straightforward manner.
3.3.1 Unitarity
Proposition 3.12 (Unitarity of Braid Elements).
Suppose , then
| (3.49) |
(As was remarked in the definition of the braids, , so equivalently, is unitary.)
Proof.
Fix , so we fix the braid elements. To prove this identity, we rely on propositions 3.3 and 3.4. Thus, we just need to show that a) and lie in the center, and b) the constant terms of and are both 1. To show that they lie in the center, we need to check that commutes with for all . Note that if or , then commutes with since it commutes with by lemma 3.8. We now note that implies the adjoint equation , which further yields by iterating the commutation relation for times. Thus, commutes with both and . Since , the only other possibilities we need to check for are or .
Recall that we have the master braid identity 3.5: . Applying this identity allows us to bring past via
| (3.50) | ||||
| (3.51) |
and past via the slightly more involved
| (3.52) | ||||
| (3.53) |
Thus, lies in the center. A similar argument using the adjoint braid identity, equation 3.6, yields the computation
| (3.54) | ||||
| (3.55) |
and
| (3.56) | ||||
| (3.57) |
so lies in the center as well.
We now need to compute the constant terms for and . A direct computation
shows that has the constant term . Similarly, has the constant term . Thus, applying proposition 3.3 in the case and , we obtain that . Similarly, again applying proposition 3.3 and setting and , we obtain that , concluding the proof.
∎
The corresponding graphical identity for unitarity, for the special case (only two generators), , is
| (3.58) |
Analogous graphical identities hold for and for general , where one puts more strands to the left and right of the above diagram. Again, we emphasize the requirement of having a diagram being represented by all strands. Hence, the above diagram does not represent the unitarity condition for all , but merely for .
In fact, we can now generalize the above unitarity condition extends to braid elements with no graphical interpretation at all:
Corollary 3.13.
| (3.59) |
for all in the set .
Proof.
Suppose without loss of generality that , and consider the isomorphism of subalgebras and given by the linear mapping satisfying , defining by its action on a basis for the subalgebra . This is an isomorphism since
, and the map is invertible. By double distributivity of multiplication in the two subalgebras, the mapping extends to a homomorphism, and thus is an isomorphism. The isomorphism maps to and to , so we obtain that . Similarly, .
∎
The above proof of proposition 3.12 may seem slightly over-kill, since we could have also expanded the product of and , and performed the double sum. The strength (and elegance) of the method becomes more apparent when one deals with more complicated products, which is what we take up next.
3.3.2 Yang-Baxter Equation and Braid Group Realization
We now give one of our main results, which is an explicit algebraic proof of a Yang-Baxter equation, using the golden rule and a systematic application of the master braid and adjoint braid identities. The Yang-Baxter equation [14] reads as and is what is known as a braid relation. More formally, we will establish the braid relations satisfied by the braid group generated by the ’s. The braid group, introduced by Artin[15], is defined to be the object
| (3.60) |
We need to show that, setting for , these ’s satisfy the relations for the braid group generators.
We first present a proof of a special case of the Yang-Baxter equation, specialized to a generalized Clifford algebra with three generators :
Proposition 3.14 (Special Case of the Yang-Baxter Equation).
| (3.61) |
Proof.
Since the braid elements are unitary, it suffices to prove the assertion that
lies in the center and
that the constant of proportionality between and is 1. By Proposition 3.4, to show that lies in the center, we just need to show that it commutes with for all . Clearly, for ,
commutes with , since each braid element commutes with . So we want to do case analysis for . For ,
| (3.62) | ||||
| (3.63) | ||||
| (3.64) |
after applying the master braid identity, Proposition 3.5 thrice and using Lemma 3.8. Applying the adjoint braid identity thrice (equation 3.6) then yields
| (3.65) | ||||
| (3.66) | ||||
| (3.67) |
as desired. The cases , are similarly shown to satisfy
| (3.68) |
in like manner. Thus, we conclude that lies in the center.
It remains to show that the constant of proportionality between and is 1. First focus on the constant terms. Since , it suffices to compare the constant terms of and . Note that in the first sum, the constant term only includes terms with and , so the constant is given by . In the second sum, the constant term only includes terms with and , so the constant is given by . Clearly the constant terms agree. However, this is not sufficient to conclude the constant of proportionality is 1, since the constant term may vanish. In fact, for , it does vanish, while it does not vanish for other . This fact is due to the following formulas corresponding to Gauss’ classical result for quadratic sums, which are tabulated in [16]:
| (3.69) |
| (3.70) |
Applying these formulas to yields that the real part of the sum vanishes if vanishes, and the imaginary part vanishes if vanishes. Thus, we require that and , so and , i.e. and , i.e. . This shows that the constant term does not vanish unless .
Now focus on the term with . In the first sum, this term is . In the second sum, this term is , so the two terms are identical. The multiplicative factor , which equals
, vanishes only for (mod 4).444I have not been able to find the corresponding Gauss sum identity in the literature, but have been able to verify this numerically using Mathematica, which shows that the half-integer-shifted quadratic Gauss sum multiplied by is periodic in mod 4.
Thus, the constant term and the term agree and their sum can never vanish. Hence, we conclude that the constant of proportionality must be 1, as desired.
∎
The corresponding graphical identity for the Yang-Baxter equation is given economically for the algebra with 3 generators , , , as
| (3.71) |
For generators, one needs to put strands to the right of the diagram for completeness.
Similar to the case of the unitarity condition, a more general Yang-Baxter-like equation holds for braid elements which do not admit a graphical interpretation:
Proposition 3.15 (General Case of the Yang-Baxter Equation).
Suppose , then
| (3.72) |
Proof.
We define an isomorphism, this time between the subalgebras and . Specifically, define by its action on a basis for the subalgebra via for all . Clearly, . Furthermore, is a homomorphism since
| (3.73) | ||||
| (3.74) | ||||
| (3.75) |
where collects all the phase factors from commuting the ’s around. It is clear that is a one-to-one mapping. Then applying to the product formula
| (3.76) |
yields
| (3.77) |
which implies the desired result by taking the adjoint braids to the other side to become braids. ∎
Now we claim that setting yields the desired braid group.
Proposition 3.16.
Set . These elements generate a unitary representation of the braid group
| (3.78) |
3.3.3 Vector Identities for the Algebraic Framework
The fact that the Yang-Baxter equation holds for the elements of the generalized Clifford algebra suggests that perhaps some kind of identities should also hold for the vectors with respect to the action of the generalized Clifford algebra. While one might speculate that the vectors (caps and cups) automatically satisfy a kind of an isotopy invariance, taking this to be a built-in axiom (in, e.g., [7]) would most certainly be incompatible with the algebraic axiomatic approach we have taken. Any such property ought to be derived from the axioms we have presented, not simply taken to be true. Of course, when working with our vectors, we must stick to the representation we have chosen for the generalized Clifford algebra, so our investigation will by necessity proceed from axiom 1 of our algebraic framework.
To those who are familiar with some subfactor theory or category theory, it may be tempting to appeal to these theories as a kind of panacea for isotopy invariance with respect to braidings. However, it must be pointed out that one cannot rely on the algebraic results of subfactor theory555Popa’s results on the axiomatization of the standard invariant [17] are for subfactors; one would need a (conjectural) graded subfactor theory, as noted in [7]. or tensor category theory666There is no tensor category here, since the tensor product is not defined between two nonneutral elements of the generalized Clifford algebra. See, e.g., [9], for a nice exposition of tensor category theory. approaches for any (we do not rule out the possibility of an explanation of the case), as these do not cover the case of parastatistics for . In fact, our algebraic framework was devised precisely to enable one to circumvent these theoretical difficulties.
As the methods of proof we developed within the algebra in the previous section cannot logically extend to proofs for the vectors, we are forced to devise new methods to prove vector identities. These methods are independent of the Yang-Baxter equation. It turns out that the results we obtain using these methods include not only graphical identities, but also encompass more general algebraic identities which supersede the graphical identities. In terms of our results, we will show that in a combinatorial sense, two basic vector identities give rise to a plethora of identifications between different vectors generated from the ground state by braidings.
First, we begin by proving a general projection-braid identity and two basic vector identities which uniformly apply to a multi-qudit space of an arbitrary number of qudits. The second vector identity, which we call the “slip” move, appears to be new. In their full generality, our two vector identities go beyond a graphical representation. We then show by example that these identities can be thought of as representing combinatorial moves that one can perform on braided states without changing the state. We conclude with an example in which we show, rigorously and without any computations, that two entangled vector states can be shown to be equal using these combinatorial moves in combination.
Thus, an important general result in this section is the introduction of a reduction procedure: in many cases, one may reduce the problem of showing equivalence of two different sequences of braidings applied to the ground state, to that of a tractable combinatorial problem, instead of one of explicit algebraic computation. The essential starting point for these vector identities is the identity lemma 2.8, and can be thought of as an important reason for using axiom 1 as an axiomatic starting point for the entire theory777 Given how the “rest” of the theory is following from the axiomatic framework, the reader perhaps is gaining more appreciation of why it was so important to separate the algebraic framework into two parts: axioms which allow one to do lots of derivations and algebraic proofs, and a proof of that these axioms are satisfied by an explicit example, i.e. the existence of a consistent vector representation of the generalized Clifford algebra that satisfied both axiom 1 and axiom 2. The division of labor is made clear, and thus each part can be independently rigorously verified..
We start with the two main combinatorial moves we will need. In this section, as a matter of form, we will draw the diagrams first, and then writing out the algebraic expressions, as the diagrams in the vector representation take on increasing importance for intuition.
Proposition 3.17 (Projection-Braid Identity, or the “Twist” Move).
| (3.79) |
Equivalently (by scaling the graphical identity by ),
| (3.80) |
More generally,
| (3.81) |
for .
Proof.
By definition,
| (3.82) |
Recall that the axioms for the projectors imply via lemma 2.8 that . So the above equality translates to
| (3.83) | ||||
| (3.84) |
The general statement follows similarly since the same lemma gives , which allows for a similar simplification from the sum over generators to a single complex number. ∎
Proposition 3.18 (“Slide” Move).
| (3.85) |
More generally (i.e. for (where is the number of strands) not necessarily equal to 2),
| (3.86) |
Proof.
Graphically, it is wisest to expand the braids on the 2nd and 3rd strands, since we may use existing algebraic graphical identities to simplify the result. This yields
| (3.87) |
Note that , commute by linear extension of lemma 3.8 so the order doesn’t matter.
In terms of a diagram, expanding the middle braids yields
| (3.88) |
where we have applied axiom 1 to bring the charge over to the 4th strand, yielding the phase factor , and then commuted it over the braid back to the 3rd strand. Similarly, the charge can be brought over the braid. Note that no additional phase accumulates, since overall the relative vertical positions of the charges are unchanged. Now apply the twist move in proposition 3.17 to get the diagram
| (3.89) |
Following the logic of the diagram, we can perform the same operations to obtain that
| (3.90) |
By unitarity of the braids, it suffices to show that .
Note that the projection onto the ground state yields by commuting past the neutral . By orthonormality of states, and equivalently, the orthonormality of states, only the terms with survive. Thus, the sum reduces to , and this is simply equal to by lemma 2.8.
Thus, it follows by unitarity of the braids that
| (3.91) |
In terms of the diagram, for , we have
| (3.92) |
∎
In terms of combinatorial moves, this identity gives us a way to “slide” one cap over the other.
Corollary 3.19.
| (3.93) |
Proof.
By taking and to the right hand side in Proposition 3.18. ∎
The above “slide” move generalizes to the general result:
Proposition 3.20 (General “Slide” Move).
| (3.94) |
for in .
Note that this result does not generally have a graphical interpretation unless .
Proof.
Again, by expansion,
| (3.95) |
The same proof as before works in this general case since we can apply the braid intertwining identities and also the twist moves (for braids and ), and then apply the axioms to simplify the vacuum expectation value. So we conclude that
| (3.96) |
∎
We would also like to be able to “slip” one cap in and out of another cap.
Proposition 3.21 (“Slip” Move).
| (3.97) |
More generally, for a positive integer not necessarily 1,
Proof.
As demonstrated in the proof of the “slide” move, this kind of proof doesn’t depend on , so long as , so let’s specialize to for convenience. The previous proposition gave a clear handle on how to manipulate the algebraic computations, so we’ll stick with the algebra.
| (3.98) |
In terms of a diagram, multiplying the state by (every cap contributes an extra factor of ) yields
| (3.99) |
since the factors of and cancel.
Undoing the twists yields factors of and , respectively, which cancel, so we are left with
| (3.100) |
Converting back to the algebraic form, one has
| (3.101) |
Note that the component has norm 1, since setting yields the component. Thus, by unitarity of the braid elements, the other basis state projections vanish, so
| (3.102) |
as desired.
∎
As with the “slide” move, there is again an algebraic generalization to braid elements with no graphical interpretation:
Proposition 3.22 (General “Slip” Move).
| (3.103) |
for in .
Proof.
By expansion,
| (3.104) |
and the same proof follows through as before. ∎
Corollary 3.23.
| (3.105) |
Proof.
By taking and to the right hand side in proposition 3.21. ∎
Proposition 3.24.
| (3.106) |
i.e.
| (3.107) |
Proof.
It suffices to show that , using the fact that .
Note that this relation does not follow immediately from the Yang-Baxter-like equation, since the Yang-Baxter-like equation does not know about the vector structure, or even about the behavior of the ground state.
First recall that proposition 3.18 says that the ground state is invariant under a
“slide” move via
| (3.108) |
and so we have that
| (3.109) |
∎
Now we prove something quite nontrivial using the above braiding relations in combination.
Proposition 3.25.
| (3.112) |
i.e.
| (3.113) |
Proof.
Equivalently, we will show that
| (3.114) |
We first substitute for following Proposition 3.18. This kills off the and braids and we are left with
| (3.115) |
Now we commute the braids which do not overlap so we get
| (3.116) |
We now substitute for to get
| (3.117) |
upon braid and adjoint braid cancellation. Now we apply the slip move in reverse to get
| (3.118) |
and then apply the slip move in reverse again to get , as desired. ∎
3.4 Significance of the Yang-Baxter Equation Proof
At this point, we wish to elaborate on the significance of our algebraic proof of the Yang-Baxter equation. This subsection is divided into two parts, the first being the particular local representation for the ’s built out of ’s satisfying the two axioms, and the second being the local representation for an alternate local representation ’s built out of ’s not conforming to the explicit representation we constructed to satisfy our two axioms, but still satisfying the relations of a generalized Clifford algebra. By local, we mean that the unitary braid elements are 2-qudit entangling gates or single-qudit gates, in the terminology of quantum circuits; and furthermore, only adjacent qudits are entangled. Via a suitable realization of the generalized Clifford algebras, the latter section provides a solution to an open question in the work of Cobanera and Ortiz [2], regarding the construction of unitary solutions realizing the braid group when the underlying qudit dimension of the -qudit system is even, of the “self-dual” form:
| (3.119) | ||||
| (3.120) |
Here, the operators and , termed Weyl generators, are defined by
| (3.121) |
and
| (3.122) |
and satisfy the commutation relation and Weyl generators with different ’s commute. The operators , correspond to the generalized Pauli operators ( is bit increment) and ( is phase increment).
3.4.1 Local Representation of the ’s
We first recall [1] the particular realization of the generalized Clifford algebras that was constructed in order to satisfy the two axioms:
| (3.123) |
and
| (3.124) |
To connect to [2], we need to rewrite and in terms of the single-qudit generalized Pauli operators, also called Heisenberg-Weyl operators. Such rewriting in terms of single-qudit operators is known as a Jordan-Wigner transformation [7]; the particular Jordan-Wigner transformation depends on some conventions about phases and the single-qudit operators chosen and needs to be computed explicitly. Thus, there was some nontriviality in verifying the axioms we presented, since we insisted on particular phases associated with the corresponding and ’s in axiom 1, which depend in some way on the parity of .
In our case, we compute the Jordan-Wigner transformation using the single-qudit operators of [2], and above. Thus,
| (3.125) |
and
| (3.126) |
First, we show that is 1-local:
Proposition 3.26.
is 1-local, i.e. it only acts on the th qudit and leaves the rest fixed. In particular, .
Proof.
| (3.127) | ||||
| (3.128) | ||||
| (3.129) | ||||
| (3.130) |
∎
It will be convenient also to have and at our disposal:
| (3.131) | ||||
| (3.132) |
Thus, the following combination is 2-local:
Proposition 3.27.
is 2-local, i.e. it only acts on the th and th qudits and leaves the rest of them fixed. In particular,
| (3.133) |
Proof.
As a consequence, we obtain the important relation that the braid elements are 2-local:
Proposition 3.28.
is 2-local. In particular,
| (3.136) |
where for each .
Proof.
Recall that
| (3.137) |
defines the braid elements. We will compute in terms of , , and .
Lemma 3.29.
Suppose , then .
Proof.
Suppose , then
| (3.138) |
. Thus, in terms of is given by
| (3.139) | ||||
| (3.140) | ||||
| (3.141) | ||||
| (3.142) |
∎
In particular, , so
| (3.143) |
Thus, applying Proposition 3.27
| (3.144) | ||||
| (3.145) | ||||
| (3.146) |
For convenience, set for each , and rewrite , yielding
| (3.147) | ||||
| (3.148) |
∎
As a consistency check, let us show that this form of the sum for is invariant under shifting the index by . The proof is nontrivial in this generalized Pauli basis, as it requires a cancellation of covariant factors. From a physics perspective, we remark that the cancellation of covariant factors is reminiscent of the construction of scalars in the theory of general relativity.
Proposition 3.30 (Cancellation of Covariant Factors).
Each term in the sum is invariant under shifting the sum index by . Thus, the sum is invariant under shifting the indexing by arbitrary integers.
Proof.
Note that if is even, since , and we can apply Lemma 3.29 for to obtain that . As , it follows that , so . Since is a primitive th root of unity, , so if is even. This is not a problem for the invariance of the sum of the braid, under shifting the index, since there are two ’s, a and a , so under shifting by , one acquires two factors of , which cancel each other out.
If is odd, the factors are invariant under shifting by since
| (3.149) |
since is an integer. Recall that in both cases, is a square root of such that so is invariant under translations by . So each term in the sum is invariant under shifting the sum index by .
Finally, it follows that shifting the indexing (e.g., from to , to to ) by arbitrary integers preserves the entire sum, since we can simply maps the terms back into by subtracting from or adding to the index of the relevant terms appropriate multiples of . ∎
It remains to compute the form of , which is accomplished with the aid of Lemma 3.29 and Proposition 3.26:
Proposition 3.31.
is 1-local. In particular,
| (3.150) |
Note that the form of the braid group generators is not in the requisite form of [2] (one may neglect the unimodular phase factor in this comparison). It is, however, sufficiently similar, if one replaces ’s by ’s, that one expects that some adaptation of our approach should work to get solutions in the form desired by [2]. We take up this problem next.
3.4.2 A General Solution to the Open Question of Cobanera and Ortiz
We now solve for braid elements of “self-dual” form given in [2]:
| (3.156) | ||||
| (3.157) |
Our construction of a realization of the braid group out of solutions of the self-dual form will depend on constructing a generalized Clifford algebra out of a particular combination of ’s and ’s. We will need to verify that the resulting particular Jordan-Wigner transformation from ’s and ’s indeed satisfies the relations of a generalized Clifford algebra. This verification step is a nontrivial point. In fact, in the original work of [2], the Jordan-Wigner transformation presented, expressing their generators and (similar to our and ’s) in terms of the ’s and ’s, is incorrect. In odd qudit dimension, they were able to use results of Jones [18] on braid group representations when is a power of an odd prime, to find a solution of the self-dual form. The flaw is that for even qudit dimension, their generators do not satisfy ! The solution, informed by our development of our algebraic framework, is to incorporate the factor of (appearing in our axiom ) to modify their Jordan-Wigner transformation. Thus, our construction illustrates once more the importance of the axiomatic approach [1] we are following, in which we both isolated the necessary algebraic structure in the two axioms, which depended on the choice of , and justified the validity of the two axioms by an explicit construction888As a reminder, is a square root of such that , which guarantees that is invariant under shifting by .. Note that since for even, can have two possible values, our construction gives rise to two distinct classes of solutions of the self-dual form.
Our starting point is Proposition 3.16, which asserts that the ’s constructed out of the generators , for , generate the braid group . Since this proof only depends on the properties of the generalized Clifford algebra, rather than on a particular representation of the algebra, the proof extends to any construction of generators out of the Weyl generators and , which satisfies the relations of the generalized Clifford algebra, namely:
| (3.158) | ||||
| (3.159) |
In the following proposition, we construct an automorphism of the generalized Clifford algebra which gives the mapping into the “self-dual” form specified by [2]. We claim that using
| (3.160) |
| (3.161) |
yields an automorphism. Since , and phases that are powers of do not affect the GCA relations, we can alternately use the mapping
| (3.162) | ||||
| (3.163) |
Proposition 3.32.
Define for by
| (3.164) | ||||
| (3.165) |
Then satisfies the relations of a generalized Clifford algebra, namely:
| (3.166) | ||||
| (3.167) |
Proof.
By Lemma 2.7, two elements of charge , where is located on generators (graphically, strands) which are left of all the generators (strands) on which is located, commute past each other with , hence for and , . So we simply need to check the commutation relation for and .
| (3.168) | ||||
| (3.169) |
Furthermore,
| (3.170) | ||||
| (3.171) |
by Lemma 3.29, where . It is clear that , hence . Thus,
| (3.172) |
Hence we have obtained an automorphism of the generalized Clifford algebra. ∎
Remark: Note that since one can construct and out of products of and and their powers and inverses, the size of the basis of the algebra is the same. This is a useful check to see whether the automorphism is actually an automorphism, independently of the relations.
Proposition 3.33 (Braid Group Representation).
Define by
| (3.173) |
where are as above. Then setting for yields a unitary representation of the braid group .
Proof.
Unitarity follows from the fact Proposition 3.12 only depends on the relations of the generalized Clifford algebra. Meanwhile, the braid group relations follow from the fact that the proof for Proposition 3.16, relying on the proof of the Yang-Baxter equation, and the commutation of elements of neutral charge, only depends on the properties of the generalized Clifford algebra as an algebra. Thus, we pass from to and Proposition 3.16 still holds. Finally, since there is freedom in the definition of the braid element by a complex phase factor, we may change to without affecting unitarity. ∎
Corollary 3.34.
More generally, by the same proof, any automorphism of the generalized Clifford algebra will preserve unitarity as well as the braid group relations.
It remains to express the ’s in terms of the Weyl generators ,.
Proposition 3.35.
is 1-local and is 2-local. They are given by
| (3.174) | ||||
| (3.175) |
Proof.
Applying Lemma 3.29:
| (3.176) | ||||
| (3.177) | ||||
| (3.178) | ||||
| (3.179) | ||||
| (3.180) | ||||
| (3.181) | ||||
| (3.182) | ||||
| (3.183) |
where we applied Proposition 3.26 to simplify .
Applying Lemma 3.29 again:
| (3.184) | ||||
| (3.185) | ||||
| (3.186) | ||||
| (3.187) | ||||
| (3.188) | ||||
| (3.189) | ||||
| (3.190) | ||||
| (3.191) | ||||
| (3.192) | ||||
| (3.193) |
∎
In the braid elements, the indexing of the coefficients and is quite curious. Partially inspired by the suggestion of Cobanera and Ortiz [2] that there may be many classes of braid group solutions of the self-dual form, we may try to extrapolate the coefficient to have different indexing. In particular, we may use the fact that the relations of the generators forming the generalized Clifford algebra are preserved under the scaling of generators and by factors of to generate different coefficients in the self-dual solutions. This appears to be related to a choice of gauge on each generator. Let us define by
| (3.194) |
where . Then the ’s again form a generalized Clifford algebra. Then the new braid elements are given by the following proposition:
Proposition 3.36.
| (3.195) | ||||
| (3.196) |
Proof.
We simply need to add in the rescaling factors induced in by the rescaling of the generators by phase factors:
| (3.197) | ||||
| (3.198) | ||||
| (3.199) | ||||
| (3.200) |
| (3.201) | ||||
| (3.202) | ||||
| (3.203) | ||||
| (3.204) |
∎
Proposition 3.37.
Setting yields a unitary braid group representation.
Proof.
The proposition follows by Corollary 3.34. ∎
Since the phase of each braid element does not affect the braid group relations, it follows that up to phase, the set of self-dual braid group solutions that we have obtained is indexed by a -dimensional vector in . Thus, using a particular automorphism of the generalized Clifford algebra and the gauge symmetry for each generator of the generalized Clifford algebra, we have obtained, from our proof of the Yang-Baxter equation and the related braid group construction, a general set of solutions to the braid group satisfying the “self-dual” form of Cobanera and Ortiz [2], which works for both odd and even ().
From a quantum computation standpoint, the braid elements are 2-local, and hence it is feasible that one might try to implement these gates. In fact, from the commutation relations 3.5 between the braid elements and the elements , and the representation of ’s in terms of the generalized Pauli operators and from equations 3.125 and 3.126, it is further evident that they almost normalize the generalized Pauli group on qudits, the almost being due to the extra factor of . To see this, simply examine the equation ; has a prefactor , but has a prefactor of , so the factor remains. Further, observe that we may recover in terms of ’s and the generalized Clifford algebra by using the expression for in terms of ’s and the expression for in terms of ’s. Thus, we can access the entire generalized Pauli group, which is generated by and ’s, by appropriate products of generators of the generalized Clifford algebra, combined with appropriate factors of ( is contained in the generalized Pauli group, so it would be redundant to keep track of factors of ). Since these products of ’s can be commuted past the braid elements to yield again products of ’s time powers of , it follows from the representation of any generalized Pauli operator as a product of generators of the algebra up to powers of that these braid elements are almost Clifford gates, where the Clifford group [19] refers to the normalizer of the generalized Pauli group within the special unitary group over qudits of dimension .
4 Explicit Computation of Some Entangled Vector States
This section is devoted to explicit algebraic computations of some entangled vector states, to demonstrate some of the variety of entangled states that can arise by braid element actions. Whereas the previous section was devoted to proof methods for showing that two vector states are equal, it did not resolve the question of what those states were, which is clearly a more complicated matter, from the computational standpoint. In proving vector identities, we were able to cleverly chain together two basic moves, the “slide” and “slip” moves, which enable one to maneuver neighboring caps over and under, as well as in and out of each other. Clearly, different methods are needed for explicit computation of the states.
In this section, we develop computational techniques which enable one to reduce vector state computation in various cases to the evaluation of a single explicit inner product, i.e. a single vacuum expectation value. Thus, the novelty here, compared with [11], for example, which also studies state computations, is that we show that state computation of entangled states using the generalized Clifford algebra is quite doable using purely algebraic methods. In fact, as we demonstrate in the final example, the braiding structures can inform one as to the strategy one should employ to reduce the state computation to the evaluation of a single explicit vacuum expectation value.
The braid elements preserve the charge of states of definite charge under the charge operator , so there is an extra symmetry. So some algebraic structure may be expected to emerge from the application of braid elements to the ground state, which is neutral.
For example, we have the following identity:
Proposition 4.1.
| (4.205) |
Proof.
By direct expansion, . As a prelude to putting the sum in normal order, we put each term into “pairwise” normal order, so . Now the action of the and elements on the ground state can be combined to yield . This is by first shifting ’s to the right of and then combining the powers of , convert the ’s to ’s via their action on the ground state.
At this point, the sum over can be explicitly evaluated since
| (4.206) |
Summing over yields (since the sum is shift invariant due to the axiom ). So we are left with , which equals as desired. ∎
Remark 4.2.
Note that if we restrict to the case of the 2-qudit ground state, then up to phase redefinition of the basis, the resulting state is of the form (as noted in [20]). More generally, we have (up to phase redefinitions) .
There is actually an easier way to get this state algebraically, using , one of the nonlocal braids we defined:
Proposition 4.3.
| (4.207) |
Proof.
Since , if we apply it to we get by bringing the charge from the fourth strand over to the third strand using the property of the ground state. Thus,
| (4.208) |
∎
We can also get rid of the extra constant factor by the following corollary:
Corollary 4.4.
| (4.209) |
Proof.
It follows from by proposition 3.17. ∎
We now compute the state given by :
Proposition 4.5.
| (4.210) |
Proof.
We give a direct computation analogous to that of proposition 4.1. Expanding all of the braids yields . Our strategy is to put all the terms in “pairwise” normal order, so we get . Using the property of the ground state under action of the ’s, we can reduce to , and then reduce to . So we are left to evaluate
| (4.211) |
which yields
| (4.212) |
which is just
| (4.213) |
as desired.
∎
As a simple example, suppose we take , so there are nine terms on the right-hand-side, yielding
| (4.214) |
Interestingly, we can write the coefficient term as , which allows us to rewrite the sum as
| (4.215) |
Following this pattern, we may conjecture that the general case is given by
| (4.216) |
Clearly, the case and hold. It turns out that this is indeed the case in general:
Proposition 4.6.
Suppose . Then
| (4.217) |
Equivalently,
| (4.218) |
Proof.
By unitarity of the braid element, it suffices to show that
| (4.219) |
whenever . The norm of the sum over these states is already 1, so this would imply that there cannot be components in addition to these neutral states.
First, observe999This series of manipulations is motivated by drawing the diagram for this vacuum expectation value, and trying to transfer the charge on the first strand over to the third strand. that we can change the to by commuting past the other ’s to act on the bra vector and then commuting back to its original position. Then we can commute past the braids until we get , which is just . This phase factor cancels the previous so we are left with the , acted on by a product of ’s and braids. We can then move past and then apply . After commuting this past the other braids we finally get
| (4.220) |
Applying this same procedure iteratively, the end result is
| (4.221) |
By assumption , so we just need to compute
| (4.222) |
Since , the only terms that contribute to the projection onto the ground state are101010This fact is justified by the axiom that the form a basis. Drawing the diagram for the expanded braid sums makes the deduction apparent. from the constant component of , and similarly, the constant component of , , etc. So we are left to evaluate
| (4.223) |
Applying the twist move times to get rid of the braids yields , so this expression evaluates to , as desired.
∎
Remark 4.7.
As seen in numerous computations for vector states, the key is to latch onto a symmetry (which may be more readily deduced from the diagram) of the vector state under the action of a neutral product of generators (which act on the vacuum state to form a basis; it is important that we project onto a basis). For a complete set of such symmetries (i.e. enough so that the square norm of the sum of projections onto the corresponding states is 1), the computation of a normalized vector state reduces to the computation of the projection onto a single vector state. Thus, in the end, only one explicit computation (expanding braid elements) must be performed.
5 Conclusion
In this work, we showed that the algebraic framework we developed in [1] allows us to construct a purely definitional graphical calculus for multi-qudit computations with the generalized Clifford algebra. Using purely algebraic methods, we established many graphical and beyond graphical identities of the representation of generalized Clifford algebras considered in the previous chapter, including a novel algebraic proof of a Yang-Baxter equation and a construction of a corresponding braid group representation. Our algebraic proof also enabled a resolution of an open problem in [2] on the construction of self-dual braid group representations for even. We also derived several new identities for the braid elements, which are key to our proofs. In terms of physics, we connected these braid identities to physics by showing the presence of a conserved charge. Furthermore, we demonstrated that in many cases, the verification of involved vector identities can be reduced to the combinatorial application of two basic vector identities. Finally, we showed how to explicitly compute various vector states in an efficient manner using algebraic methods.
Furthermore, we demonstrated that it is feasible to envision implementing the braid operators for quantum computation, by showing that they are 2-local operators. In fact, as we demonstrated these braid elements are almost Clifford gates, for they normalize the generalized Pauli group up to an extra factor .
Acknowledgments
I would like to express my gratitude and thanks to Professor Arthur Jaffe, for helpful discussion and guidance, especially his recommendation to find a more general identity for the braid.
I have been supported in the later stages of this work by ARO Grant W911NF-20-1-0082 through the MURI project “Toward Mathematical Intelligence and Certifiable Automated Reasoning: From Theoretical Foundations to Experimental Realization.”
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