How hard is it to approximate the Jones polynomial?

Freedman, Kitaev, and Wang [arXiv:quant-ph/0001071], and later Aharonov, Jones, and Landau [arXiv:quant-ph/0511096], established a quantum algorithm to"additively"approximate the Jones polynomial V(L,t) at any principal root of unity t. The strength of this additive approximation depends exponentially on the bridge number of the link presentation. Freedman, Larsen, and Wang [arXiv:math/0103200] established that the approximation is universal for quantum computation at a non-lattice, principal root of unity; and Aharonov and Arad [arXiv:quant-ph/0605181] established a uniform version of this result. In this article, we show that any value-dependent approximation of the Jones polynomial at these non-lattice roots of unity is #P-hard. If given the power to decide whether |V(L,t)|>a or |V(L,t)|b>0, there is a polynomial-time algorithm to exactly count the solutions to arbitrary combinatorial equations. In our argument, the result follows fairly directly from the universality result and Aaronson's theorem that PostBQP = PP [arXiv:quant-ph/0412187].


INTRODUCTION
A well-known paper of Aharonov, Jones, and Landau [4] establishes a polynomial quantum algorithm to approximate the Jones polynomial at any principal root of unity; a more abstract form of this algorithm appeared previously in a paper of Freedman, Kitaev, and Wang [10]. Theorem 1.1 (Freedman,Kitaev,Wang [10]; Aharonov, Jones, Landau [4]). Let t = exp(2πi/r), let L be a link presented by a plat diagram with bridge number b, and let V (L,t) be its Jones polynomial. Then there is a polynomial-time quantum decision algorithm that answers yes with probability (In the AJL version of the result, the algorithm is jointly polynomial time in the r, the order of the root of unity; as well as in the bridge number and the crossing number. They also refine the algorithm to estimate V (L,t) as a complex number rather than just estimating its length.) Aharonov, Jones, and Landau describe the error in this algorithm as additive, and note that it would be much harder to provide algorithm with multiplicative error. Multiplicative error would mean that V (L,t) or |V (L,t)| can be approximated to within a constant factor of its value.
Another way to distinguish between types of error is to say that the error in the approximation algorithm is presentationdependent. Given different plat presentations of the same link L, the error grows exponentially in one of the parameters of the presentation, namely the bridge number. An algorithm to approximate the Jones polynomial is only directly useful for topology if its error is value-dependent, i.e., independent of * Electronic address: greg@math.ucdavis.edu; This material is based upon work supported by the National Science Foundation under Grant No. 0606795 quantities other than the value of |V (L,t)|. Multiplicative error is one type of value-dependent error, but in some cases valuedependent error is more general. (Multiplicative error is used to define complexity classes such as APX; see Section 2.1. ) Freedman, Larsen, and Wang [12] established that the approximated quantity V (L,t) (t 1/2 +t −1/2 ) b−1 2 in Theorem 1.1 is universal for quantum computation when r = 5 or r ≥ 7. (Aharonov and Arad [2] establish an r-uniform version of this result.) Thus, even if the error is presentation-dependent, the approximation is computationally valuable for carefully chosen link diagrams.
On the discouraging side, Jaeger, Vertigan, and Welsh [17] showed that it is #P-hard to exactly compute the Jones polynomial V (L,t) except when t is a root of unity of order r ∈ {1, 2, 3, 4, 6}. They established a reduction from the Tutte polynomial of a planar graph to the Jones polynomial of an associated link.
The main result of this article is that the "encouraging" universality result strengthens the "discouraging" hardness result: Any value-dependent approximation of above values of the Jones polynomial is #P-hard. The argument is a mashup of three standard theorems in quantum computation: The Solovay-Kitaev theorem [20] (although only a mock version is needed), the FLW density theorem, and Aaronson's theorem that PostBQP = PP [1]. (See also [8] for a different hardness result.) Theorem 1.2. Let I(L) be a quantumly finite, unitary link invariant which is eventually dense. In particular, I(L) can be the Jones polynomial V (L,t) with t = exp(2πi/r) and r = 5 or r ≥ 7. Let a > b > 0 be two fixed positive real numbers, and assume as a promise that |I(L)| ≥ a or |I(L)| ≤ b. Then it is #P-hard, in the complexity of the diagram of L, to decide which inequality holds; moreover we can take L to be a knot.
The idea of Theorem 1.2 applies to many cases where the invariant is not unitary; and when it is not an invariant of links but some other type of topological or combinatorial object. We have no formal statement of a general result, but the basic argument is that if an invariant can simulate a quantum computer, then typically multiplicative or value-dependent ap-proximation is universal for PostBQP and therefore #P-hard.
Here is an example result of this type, modulo a denseness result which is explained in Section 4 and which will be established in a later paper. (We also provide a Sage program to confirm an adequate weak form of the conjecture for specific values of n.) Theorem 1.3. Multiplicative approximation to the Tutte polynomial T (G, x, y) for planar graphs G is #P-hard for any fixed values of x and y such that n = (x − 1)(y − 1) ≥ 4 and x, y < 0, except x = y = −1. Multiplicative approximation of the Jones polynomial is also #P-hard for any real t > 0 other than t = 1.
In related results, Goldberg and Jerrum [14] showed that multiplicative approximation of many values of the Tutte polynomial T (G, x, y) is NP-hard (relative to RP) for nonplanar graphs, while some values are #P-hard. Jaeger, Vertigan, and Welsh [17] also analyzed when T (G, x, y) is #P-hard to compute exactly. They noted that the Jones polynomial V (L,t) of an alternating link L is equivalent to T (G, x, y) for a planar graph G along the curve xy = 1. More recently [13], Goldberg and Jerrum also established that many values of the planar Tutte polynomial are NP-hard to approximate. Their new theorems have some intersection with Theorem 1.3, but their constructions are very different.

Acknowledgments
The author would like to thank Scott Aaronson, Dorit Aharonov, Leslie Ann Goldberg, and Eric Rowell for useful remarks.

COMPLEXITY THEORY
We assume that the reader is somewhat familiar with complexity classes such as P, NP, BQP, #P, and the notation that A B means the class A with oracle B. See the Complexity Zoo [25] and Nielsen and Chuang [20] for a review.
Whereas a problem in the class #P counts the number of witnesses accepted by a verifier in polynomial time, and a problem the class NP reports whether there is an accepted witness, a problem in the class PP reports whether a majority of the witnesses are accepted. A problem is #P-hard if and only if it is PP-hard, because P PP = P #P (exercise for the reader). On the other hand, a problem which is #P-hard is also hard for the polynomial hierarchy PH, by the deeper theorem due to Toda [23] that The class NP with a tower of n − 1 NPs as an oracle is called the nth level of the polynomial hierarchy. One of the standard conjectures in complexity theory is the polynomial hierarchy does not collapse, i.e., that nth level does not equal the n + 1st level for any n. Thus by Toda's theorem, if a problem is #P-hard, then it is viewed as qualitatively harder than if it is merely NP-hard.
The class PostBQP is defined as polynomial-time quantum computation with free retries, or postselection. In other words, the computation can output "yes", "no", or "try again", and the probability of "yes" is required to be c times that of "no" or vice-versa for some constant c > 1. All values of c are equivalent because c can be amplified by repeated trials. There is an analogous class PostBPP for classical randomized computations; it was also defined previously as BPP path . Aaronson [1] established that PostBQP = PP. It is not hard to show that PostBQP is a subset of PP, just as BQP, NP, and a number of other important classes are. The more surprising fact is that PostBQP is all of PP.
By contrast, PostBPP is unlikely to be all of PP. The relevant complexity results are as follows: 1. PostBPP contains P ||NP (P with parallel NP queries) [15].
Thus, PostBPP is known to be in the third level of PH and it is presumably in the second level. Another interpretation of PostBQP or PostBPP is that it consists of two computations performed by Alice and Bob. Given a decision problem L, Alice and Bob each either succeed or fail in separate polynomial time with probabilities a and b. If L(x) is "yes", then a > cb, while if L(x) is "no", then b > ca. This description already reveals the main idea of Theorem 1.2: If we can estimate both a and b with error less than c, then we can compare them up to a factor of c 2 .
It is interesting and relevant to compare PostBQP and PostBPP to three other complexity classes: A 0 PP (one-sided almost wide PP), SBP (small-bounded probabilistic P [5]), and what we will call SBQP. All three classes depend on a real-valued function f (x) in FP (expressed in fixed-point arithmetic, say), where x is the input to the decision problem, and a constant c > 1. The classes SBP and SBQP are defined in exactly the same way as PostBPP and PostBQP, except that Bob's value b is not a success probability but rather computed directly in FP (as a real number in fixed-point arithmetic, say). Meanwhile A 0 PP is defined like SBP, in which a is a success probability and b is computed in FP, except with the formulas Proof. The constant c is irrelevant by the usual technique of amplification by repeated trials. This is immediate in the case of SBP and SBQP. It is not very difficult in the case of A 0 PP, and was established by Vyalyi [24].
To argue that f (x) can be set to 2 −p(|x|) (in the cases of SBP and SBQP), first choose p so that f (x) > 2 −p(|x|) . Then Alice can compute f (x) and reduce her success probability by a factor of 2 p(|x|) f (x). The argument in the case of A 0 PP is essentially the same and was also explained by Vyalyi [24].
Proof. The proof is almost the same as Aaronson's proof that PostBQP = PP. We can also define A 0 PP as a counting class in which, for each certificate y of length n, the computation produces a value f (y) = ±1, and these values are summed to produce A(x). For a decision problem L ∈ A 0 PP, we require that First, let L ∈ SBQP be computed by a quantum circuit that consists of Hadamard and Toffoli gates. It is convenient to change the counting model of A 0 PP slightly to let the values be ±1 or 0. Then we obtain an A 0 PP algorithm by multilinear expansion of the effect of these gates on density matrices. The matrix entries of a Toffoli gate, in its effect on a density matrix, are 0 and 1; the corresponding matrix entries of a Hadamard gate are ± 1 2 . The final probability is given by a partial trace of the output density matrix, and is non-negative and exactly matches the criteria for A 0 PP. Now let L ∈ A 0 PP and let a be Alice's success probability in the A 0 PP algorithm. We can again slightly re-express the counting model of A 0 PP so that f (y) ∈ {0, 1} and its sum Then, in the SBQP algorithm, we can quantum-compute the unitary map where the value f (y) is written to an ancilla qubit. We provide the input | + + · · · + to U f , and then postselect on whether the left n qubits of the result are all |+ . If they are, then the ancilla qubit has the state If this qubit is measured in the ± basis, then the probability of |− is If we assume that b > 1 4 and let C = 2 be the threshold factor in the A 0 PP algorithm, then So we can let b ′ = 4b 2 and C ′ = 2 in an SBQP algorithm that produces the probability a ′ .
Many of the complexity classes discussed here employ the semantic condition that the probabilities of particular outcomes are above or below certain thresholds. We can also consider promise versions of these classes in which this threshold gap may only present for some inputs and not others. When they are considered in promise form, SBQP and SBP become equivalent for our purposes to PostBQP and PostBPP.
Proof. Suppose that L ∈ PostBQP (say) and that it is implemented by a quantum circuit. We assume that the gates of the circuit are Hadamard and Toffoli gates (see the remark below), so that the probabilities a and b, if they are non-zero, are bounded below by 2 −O(g) in a circuit with g gates. They are therefore also bounded below by 2 −poly(|x|) , where x is the input.
The construction is then similar to another part of Aaronson's proof that PostBQP = PP. We assume that either a > 8b or that b > 8a. Choose an n, no more than polynomial in |x|, such that max(a, b) > 2 −n . Then for each 0 ≤ k ≤ n, use PromiseSBQP to compare both a and b to 2 −k . If a > 8b, then for every k, PromiseSBQP will either reliably report that a > 2 −k or that 2 −k > b, and there will exist a k for which it will do both. Meanwhile if b > 8a, it will report that b > 2 −k or that 2 −k > a, and both for at least one k. These two outcomes are mutually exclusive.
The argument that PostBPP ⊆ P PromiseSBP is the same, but simpler since the lower bound on max(a, b) is immediate.
Remark. Gate-set independence is a non-trivial property of PostBQP. Because the probabilities a and b are polynomials in the matrix entries of the gates, and using norm bounds on the coefficients of those polynomials, the gates can be varied slightly so that a, b > 2 −poly(g) . The Solovay-Kitaev theorem can then be applied to show that PostBQP is a gate-independent complexity class. By contrast, we do not know whether post-selected quantum computation is gateindependent with a time bound ofÕ(n α ) for some fixed exponent α, because the Solovay-Kitaev theorem could change the exponent.
Finally, as noted by Aaronson, linear computation is another interesting interpretation of PostBQP.
Postconditioning allows us to replace unitary gates by subunitary gates, and to rescale subunitary gates arbitrarily. But every linear operator that acts on vector states |ψ is proportional to a subunitary operator. Thus, PostBQP can also be defined by the class of polynomial-sized circuits with linear gates, without the unitary restriction. It may seem then as if PostBQP uses the state space of a qubit only as a 2-dimensional complex vector space, and not as a Hilbert space. However, the definition of measurement probabilities in PostBQP still use the Hilbert space structure.

Approximation classes
The approximation classes listed in the Complexity Zoo [25] that express multiplicative error include APX, PTAS, and FPTAS. These classes are defined there for optimization problems, but they can equally well be defined for arbitrary functional problems. Let f (x) be a function f (x) that takes strings x to positive real numbers. Then f (x) is in APX if it can be approximated to within some bounded factor in polynomial time (with fixed-point output); it is in PTAS if it can be approximated to within a factor 1 + ε in polynomial time for any ε > 0; and it is in FPTAS if the computation is jointly polynomial time in |x| and 1/ε. (These classes all refer to deterministic computation; there are analogous randomized classes such as FPRAS.) We do not know of a standard complexity class to express general value-dependent error, so we define one here. Let f (x) be as before. Then f is in APV if for every constant a > 0, there exists a constant b > a and a polynomial-time algorithm to decide whether f (x) > b or f (x) < a, given the promise that one of the two is true. Similarly, we could define a randomized version ARV; and both APV and ARV have a variation which is uniform in the choice of the constant a.
The following proposition says that if f (x) can be suitably rescaled, then general value-dependent error becomes equivalent to multiplicative error. The proposition and its proof use the same idea as the proof of Proposition 2.3, and a similar rescaling argument in Aaronson's argument that PP ⊆ PostBQP.

Proposition 2.4. Suppose that f (x) takes positive real values
and is in APV, and suppose further that | log f (x)| is bounded by a polynomial in the length |x| of the input. Suppose that there are constants c > 1 and k > 1 such that every integer n, there is a reduction y n (x) such that and suppose that this reduction can be computed in joint polynomial time in n and in |x|. Then f (x) is in APX.
Proof. (Sketch) Let a and b be some constants such that we can decide by a subroutine whether f (x) < a or f (x) > b in polynomial time. Then we can bound f (x) to within a factor of cb/a. We know by hypothesis that f (x) > k −m and f (x) < k m for some m which is polynomial in |x|. So the strategy is to ask whether f (y n (x)) is more than a or less than b for every |n| ≤ m. The largest n for which the subroutine reports that f (y n (x)) < a yields a good estimate of ak −n . The estimate is within a factor of cb/a, even though the subroutine could give a false yes answer when f (y n (x)) < b.

QUANTUM INVARIANTS
Let k be a field and let I(L) be a k-valued invariant of links. Assume that I(L) is multiplicative, meaning that if L = L 1 ∪L 2 consists of two links separated by a plane, then Depending on the nature of I, the link L could optionally have other decorations such as a framing, a coloring, or an orientation. (An orientation refers to orienting each component of the link as a curve; ambient R 3 is always oriented by convention.) Now choose a fixed plane R 2 ⊆ R 3 and consider those links L = A ∪ B that are formed from two tangles A and B. Then I can also be seen as an invariant I(A, B) of the two tangles, defined when A and B have matching boundary Σ so that they can be glued together to make a link:

A B
We (For related constructions, see also [7] and [9].) A linear relation among tangles in H Σ , whether or not it is finite-dimensional, is called a skein relation. The space H Σ itself is called a state space or a skein space. Also, the multiplicative property or L implies that if Σ is divided into two disjoint parts, Σ = Σ 1 ∪ Σ 2 , then The state space H Σ is also naturally a representation of the braid group of Σ. In the case of the Kauffman bracket, this can be taken to be the usual braid group B n , but in general it will be a braid group that preserves the colors or orientations of the points in Σ.
If A and B are two matching tangles as before, let A and B be their reflections through their separating plane. If the field k = C and I conjugates under reflection, then

I(B, A) = I(A, B).
In this case I (A, B) induces a non-degenerate Hermitian form on H Σ . If this form is positive-definite, then H Σ is naturally a Hilbert space and the invariant I is unitary. If I is unitary, then we can also write it and the associated tangle invariant in Dirac notation: Say that a unitary link invariant I is fully non-trivial if it satisfies the following two conditions.

2.
If u c is an unknot with any color c, then u c I > 1.
In fact, the second condition follows from the first one for unitary invariants, but it is simpler for us to take both conditions as the definition. Note in particular that if I is fully non-trivial, then dim H Σ ≥ 2.
An important example of these axioms is the Kauffman bracket form of the Jones polynomial. The bracket is defined for unoriented links with "blackboard framing", meaning that the link is a ribbon laid flat along the link diagram. If t is the indeterminate of the Jones polynomial, then the defining skein relations are (Note also that the Jones polynomial is often also parameterized by the variable q = t Here w(L) is the writhe of the diagram of L, i.e., the number of positive crossings minus the number of negative crossings.
(The Jones polynomial is also quantumly finite after rescaling to make it multiplicative, but the Kauffman bracket formulation is generally more convenient.) If I(L) is a quantumly finite invariant of links, then there is an associated action of a braid group on each skein space H Σ . If Σ has n equivalent marks, as in the case of the Kauffman bracket, then we can take the braid group to be the standard braid group B n . (In more general cases, the braid group has to preserve the colors or orientations of the marks on Σ.) If I(L) is unitary, then we say that it is eventually dense if the image of the braid group is dense in PSU(H Σ ) when Σ has enough points (of every available color, if there is more than one). Theorem 3.1 (Freedman, Larsen, Wang [12]). The Kauffman bracket is eventually dense if t = e 2πi/r with r = 5 or r ≥ 7.
The following proposition was first argued by Freedman, Larsen, and Wang in a special case [11]. Our argument is not really different from the standard one, but we give it for completeness. For simplicity, we assume that I is an uncolored link invariant in the rest of the section.

Proposition 3.2. Suppose that a quantumly finite, fully nontrivial link invariant I is eventually dense and that for each fixed L, the value of L I has an FPTAS. Let p(x) be the probability that some quantum-polynomial time algorithm accepts an input x. Then the input x can be encoded as a link L = L(x) with bridge number g, so that
where "≈" is in the FPTAS sense and u is an unknot.
The condition that L I has a FPTAS for each L is fairly mild. It says that L I can be approximated to within ε in polynomial time in 1/ε, and that the polynomial can get worse as L gets bigger. In the proof this estimation property will be combined with the Solovay-Kitaev theorem. However, the Solovay-Kitaev theorem is much stronger than necessary: it establishes that quantum gates can be approximated with words of size polynomial in log ε, and that the words can be found in the same time complexity.
Proof. There is more than one way to argue Proposition 3.2; in this article we will use the method of plat presentation. For simplicity we will describe the case of an uncolored, unoriented link invariant; the more general case works the same way.
A plat presentation of a link L is a diagram that consists of a braid B on 2g strands, with n copies of the cap v at one end and g copies of the cup v at the other end: The number g is the bridge number of the plat presentation.
First, using uncomputation, we can make a quantum circuit C = C(x) so that the acceptance probability is | 0 n |C|0 n | 2 , and such that both the gate number |C| and the qubit number n are polynomials in |x|.
Second, by hypothesis the space V (4) for the invariant I is at least 2-dimensional. We can use a 2-dimensional subspace of V (4) as a computational qubit. More precisely, we choose this subspace to include the value of 2v, where v is a cap:

v =
The computational state |0 is defined to be proportional to |2v in the computation. Since I is eventually dense, there exists an n such that the closure of the braid action on V (4n) includes 2-qubit gates acting on V (8). Thus the circuit C can be approximated by a braid B on 4n strands. The FPTAS hypothesis implies that the matrix entries of the braid generators can be approximated in polynomial time, and the Solovay-Kitaev theorem implies that B can be generated as an FPTAS of C.
Let L be the link which is the plat closure of B. Then where as usual u is an unknot. We let g = 2n. Since this norm of the left side is the acceptance probability, this completes the proof of the proposition.
In light of the analysis in Section 2 of PostBQP, Proposition 3.2 establishes that if I(L) is unitary and eventually dense, then a multiplicative approximation to it is PostBQP-hard. By Aaronson's theorem, it is therefore also #P-hard.
One remaining task to prove Theorem 1.2 is to refine the construction to general value-dependent approximation rather than multiplicative approximation. Explicitly, let a > b > 0 be constants as in the statement of Theorem 1.2, let p and c be the polynomial the constant in the modified definition of SBQP in Proposition 2.1. By that proposition and equation (1), it is SBQP-complete and therefore #P-hard to determine whether is more than c2 −p(|x|) or less than 2 −p(|x|) . In other words, it is #P-hard to determine whether We want to make a modified link L ′ to make it hard to determine whether | L ′ | 2 is more than a or less than b. Recall that b is a polynomial in |x| and that | u I | > 1. If when |x| is large, then we can add m(|x|) copies of the unknot to L so that | u I | g+m 2 −p is bounded. On the other hand, if 2(log 2 | u I |)n(|x|) ≫ p(|x|) then we can use denseness to first create a link L 0 (say a 2bridge link corresponding to a 1-qubit circuit) such that | L 0 | is a small constant. Then we can add m(|x|) copies of L 0 to L so that is bounded. The constant c in the definition of SBQP can be chosen to overwhelm the bound in either case as well as the specific values of a and b.
Finally we want to show that L can be a knot. The trick for this is that since the braid group is eventually dense, the pure braid group is also eventually dense. Thus in the braid B we can switch two strands, and then approximately cancel its effect with a pure braid that does not permute any strands. The permutation induced by B is thus decoupled from the approximate value of L I , so it can be chosen so that L has only one component. This completes the proof of Theorem 1.2.

THE TUTTE POLYNOMIAL
The aim of this section is to prove Theorem 1.3. In order to understand the Tutte polynomial, we will first convert it to another graph invariant with equivalent information known as the Potts model. The Potts model of a graph G depends on a positive integer n, the number of colors; and on a variable y. The weight of a coloring of the vertices of G with n colors is defined as y k if k of the edges of G connect two vertices of the same color. Then the Potts partition function Z(G; n, y) is defined as the total weight of all vertex colorings. The Potts partition function yields the Tutte polynomial T (G; x, y) by the formula where n = (x − 1)(y − 1). A more standard definition of the Tutte polynomial is via a contraction-deletion formula. However, the definition using the Potts model is easily equivalent.
In order for the complexity of a value of the Tutte polynomial or Potts model to be well-defined, the parameters themselves must be well-defined computationally. One conservative choice is to let x, y, and n be rational numbers. However, for our purposes we can let x, y, and n be any real numbers that are themselves defined by an FPTAS. (Or complex numbers, although this section has no bearing on them.) We can define a graph with boundary to be any graph G together with a marked subset ∂ G of the vertices, by definition the boundary vertices. Given two graphs A and B with boundary and a bijection between ∂ A and ∂ B, we can glue them together to make a graph G, in the same way that two tangles can be glued together to make a link. Thus for any invariant I(G) of graphs, we can also define the relative invariant I(A) and say what it means for I to be quantumly finite. There is a state space V (k) for each number of vertices k. We can also consider the same constructions for planar graphs; one difference is that the boundary vertices ∂ A of a planar graph A must all lie on the outside face of A.
Although we could define the Potts model Z and the state space V (k) over any field, in this section we will work over the real numbers R. Here is an example of a shrub forest, in this case a planar one, which can be used as a basis vector for V (k): Proof. (Sketch) This is a folklore result in enumerative combinatorics and quantum algebra. For each partition P of the boundary k vertices, we can construct a shrub forest such that each parts of P is the set of leaves of one of the shrubs. In the planar case, we can consider planar shrub forests and planar partitions. In both cases, the shrub forests are a basis of V (K).
The fact that shrub forests, respectively planar shrub forests, span follows from the contraction-deletion formula for the Potts model. The fact that they are linearly independent can be established by writing down the matrix I (A, B), where A and B are two shrub forests. For each entry of this matrix, look at the leading term as a polynomial in y, and then look at the leading term in n of its coefficient. It is not hard to show that this leading term of the diagonal term of detI(A, B) cannot be cancelled, so that I (A, B) is non-singular.
An important variation of the Potts model (or the Tutte polynomial) is the multivariate version, where the weight y can be different for each edge of G. It will be important to assign each edge not only its weight y, but also the dual weight x using the relation n = (y − 1)(x − 1). (Note that there is only one value of n for all of G.) The Potts model admits a renormalization procedure so that the ordinary Potts model becomes equivalent, for the purpose of #P-hardness of multiplicative approximation, to part of the multivariate Potts model [14]. If G has two parallel edges with weight y 1 and y 2 , then they are equivalent to a single edge with weight y 1 y 2 . Meanwhile, if G has two edges in series with dual weight x 1 and x 2 , they are equivalent (up to changing Z by a constant factor) to one edge with weight x 1 x 2 . These transformations are called shift operations; they are also called compositions and implemented weights [13]. Note that the series and parallel compositions also preserve planarity.
The point of implemented weights is the following reduction: Proof. (Sketch) First, for any n > 1, suppose that we can obtain some weight y ′ , and let x ′ be the dual weight. Then by taking powers of the dual weight x ′ , we can obtain weights y ′′ that converge to 1. By taking powers of these weights, we can obtain a dense set of weights y ′′′ > 1. With more careful accounting, this strategy converges to any weight y ′′′ > 1 with polynomial overhead. Now suppose that n > 2 and that y > −1. Then y ′ = y 2 > 1, so by the previous argument, we can obtain all weights y ′′ > 1. We can now multiply these weights by the original weight y, and the dual weights x ′′ by the dual weight x, to obtain all weights y ′′′ < 1. Likewise if x > −1, we can also obtain all weights.
Suppose instead that x, y ≥ −1. By hypothesis, either 0 > x > −1 or 0 > y > −1; suppose the latter without loss of generality. Then we can let y ′ = y n for a large enough odd power n, and obtain x ′ < −1. Thus we can reduce to the previous case.
The following diagram shows the different cases for two values of n. We can define two types of operators that act on the state space V (k) and that generate a semigroup which is analogous to the braid group action on the state space of the Jones polynomial in Section 3. The operators are expressed by graphs with two subsets of vertices, which may intersect, and which are called left and right boundary. One type of operator A y is given by connecting two of the vertices which are adjacent with an edge with weight y; all k of the vertices are both left and right vertices. The other type of operator B x is given by replacing one of the k vertices by a vertical edge with dual weight x (denoted with parentheses): In this case the other k − 1 vertices are upper and lower boundary, while the edge connects an upper vertex to a lower vertex. There are k positions for both of the operators, to make operators A j,y and B j,x for 1 ≤ j ≤ k; call them edge operators.
In general, a set of edge operators generates a semigroup rather than a group. However, the calculus of compositions implies that j,x . Therefore: Proposition 4.4. Consider the edge operators A j,y and B j,x generated by a set of weights Y and the associated dual weights X. If X and Y are both closed with respect to taking reciprocals, then these edge operators generate a subgroup of the projectivized linear group PSL(V (k)).
We will also need the following theorem.
Theorem 4.5. The edge operators A j,y and B j,x , using all x, y = 1, generate PSL(V (k)) (over R) when n ≥ 4.
We will prove Theorem 4.5 in a later paper [19]. In the meantime, we include a Sage code that can establish Theorem 4.5 for k = 8 and for any specific numerical n [18]. By the proof of Proposition 3.2, this is enough to establish that the model is universal for linear computation and therefore #Phard to estimate. See also Aharonov, Arad, Eban, and Landau for closely related results [3].
By analogy with Proposition 3.2, Proposition 4.3 and Theorem 4.5 together imply that linear circuits can be encoded as graphs in the Potts model when n ≥ 4 and y = 1, in the same way that quantum circuits can be encoded as links using the Jones polynomial at a root of unity.
Thus, Aaronson's theorem again implies that multiplicative approximation of the Potts value Z(G, y, n) is #P-hard when n ≥ 4, y < 1, and G is planar. This is the first part of Theorem 1.3.
For the second part of Theorem 1.3, it has long been known [17] that the Jones polynomial V (L,t) of alternating links L is equivalent up to normalization to the Potts model Z(G, x, y) or the Tutte polynomial with x = −t and y = −t −1 (or vice versa) for planar graphs G. One argument for this is that by replacing each edge of G with a crossing, we can convert a contraction-deletion formula for the Potts model to the Kauffman relations. Since then n = t + 2 + t −1 , we can conclude that multiplicative approximation of the Jones polynomial is #P-hard when t is real and positive and t = 1. Remark. The Jones polynomial for arbitrary links is equivalent to a certain multivariate, planar Potts model, and the state spaces of the Jones polynomial are isomorphic to those of the Potts model. We have already stated that the dimension of either state space is a Catalan number.

Approximation with extra information
Theorem 1.2 says that value-independent approximation of certain values of the Jones polynomial are #P-hard even when the link L is taken to be a knot. We conjecture that L could in addition be a prime knot or even an atoroidal knot. Maybe other such restrictions on the structure of L could be imposed. But without a result such as that distinguishing the unknot (say) is hard, it is not feasible to add arbitrary interesting topological restrictions on L to Theorem 1.2. Maybe there is a fast algorithm to distinguish the unknot. The Jones polynomial would then be easy to compute for knots that are recognized as the unknot or as some other specific knot.

Other kinds of approximation
There are many other kinds of partial information about the Jones polynomial without any interesting complexity bound to our knowledge. Is the degree of the Jones polynomial intractable? Is it intractable to determine when the Jones polynomial vanishes? What if the Jones polynomial is reduced mod p for some prime p?

Denseness may be more than necessary
It is easiest to see that a set of gates is universal for linear computation if they densely generate an appropriate Lie group. For instance, they might generate PSL(2 n , C) if they act on n qubits, or PSL(2 n , R) inside it. However, dense generation is more than necessary for certain types of universality. For example, 2-qubit gates that are in PSL(4, Z) do not densely generate any Lie group. But suppose that the gate which is proportional to a rotation by an irrational angle, and which does densely generate PSL(4, R) together with other integer gates. We do not know the right criteria on linear gates to establish #P-hardness results.

Morse algorithms may be optimal
It is common practice to compute the Jones polynomial by a strategy known variously as a Morse algorithm, dynamic programming, a scanline algorithm, or a divide-and-conquer algorithm. (Morse theory in geometric topology is a theory of analyzing a topological object by dividing it into horizontal slices.) For a knot in a plat presentation, the strategy is to numerically compute the action of the braid group on the skein space. This type of algorithm requires simple exponential time and space in the number of strands of the braid, or for other kinds of knot diagrams, the width of the diagram. This is much better than a direct recursive evaluation of the Jones polynomial using a finite set of skein relations; the time complexity of any such direct algorithm is instead exponential in the number of crossings.
It is natural to wonder whether there are other clever algorithms that can compute the Jones polynomial even faster. However, the proof of Theorem 1.2 could be evidence that Morse algorithms are essentially optimal for many kinds of knot diagrams. In short, if braids are evaluated using the Jones polynomial at the dense roots of unity of Theorem 1.2, then they are a model of general planar quantum circuits.
In more detail, consider a typical hard search problem based on classical circuits, and an analogous problem based on quantum circuits. For instance, let (z, w) = C(x, y) be a reversible circuit whose input (x, y) and output (z, w) are each divided into two registers of equal length. Then it is NP-hard to determine whether there is a solution to (z, 0) = C(x, 0). We conjecture that there are linear-depth, planar circuits C for which this problem requires exponential time in |x|.
Moreover, using denseness at a convenient root of unity and the Solovay-Kitaev theorem, this circuit problem can be encoded in a braid with polylogarithmic overhead. We conjecture that extra logarithmic factors are not necessary for hardness. We conjecture that there exist linear-length plat presentations of knots, such that the Jones polynomial requires exponential time in the bridge number b to estimate at a dense root of unity. Such conjectures are very difficult to prove unconditionally, because they would imply that #P is not contained in FP. But if they were accepted, then Morse algorithms would be accepted as essentially optimal.