BQP-Complete Problems

Abstract

The concept of completeness is one of the most important notions in theoretical computer science. PromiseBQP-complete problems are those in PromiseBQP to which all other PromiseBQP problems can be reduced in classically probabilistic polynomial time. Studies of PromiseBQP-complete problems can deepen our understanding of both the power and limitation of efficient quantum computation. In this chapter we give a review of known PromiseBQP-complete problems, including various problems related to the eigenvalues of sparse Hamiltonians and problems about additive approximation of Jones polynomials and Tutte polynomials.

These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

1 Introduction

A celebrated discovery in theoretical computer science is the existence of NP-complete problems. A decision problem is NP-complete if it is in NP, and all other problems in NP reduce to it in deterministic polynomial time. After Cook (1971) (and Levin (1973) independently) showed that the Satisfiability problem is NP-complete, Karp (1972) found that 21 natural combinatorial problems, mostly problems on graphs, are NP-complete as well. Since then, thousands of problems arising from various disciplines such as mathematics, physics, chemistry, biology, information science, etc. have been found to be NP-complete (Garey and Johnson 1979).

Why is the notion of NP-completeness so important? First, showing a problem to be NP-complete means that the problem is very unlikely to be solvable in polynomial time. This makes NP-completeness an “important intellectual export of computer science to other disciplines” (Papadimitriou 1997). Second, the number of NP-complete problems that arise naturally is huge; thus NP-complete problems give a nice classification of NP problems. Actually, all but a few NP problems are known to be either in P or NP-complete. Nowadays, whenever people find a new problem in NP and cannot quickly find a polynomial-time algorithm, they start to consider whether it is NP-complete. Third, the notion of completeness provides a useful method to study a whole class by concentrating on one specific problem. For example, if one wants to separate NP from P, it is enough to show that any NP-complete problem does not have a deterministic polynomial-time algorithm. Of course, if one tries to explore the possibility of NP = P by designing efficient algorithms for NP, it is also sufficient to find an efficient algorithm on any one of the NP-complete problems.

Complete problems do not only exist in NP, they also exist in other computational classes, although the requirement for the reduction algorithm may vary from case to case. To study a property of a class, it is enough to show the property for one complete problem in the class, as long as the reduction used in the completeness definition does not destroy the property. For example, the fact that Graph 3-Coloring has a (computational) Zero-Knowledge protocol immediately implies that all NP problems have a Zero-Knowledge protocol, because for an arbitrary NP problem, the prover and verifier can first map the input to a Graph 3-Coloring instance and then run the Zero-Knowledge protocol for the latter (Goldreich et al. 1991).

Quantum computing is a new paradigm rapidly developed since the mid-1990s. Since Shor's fast quantum algorithm for Factoring and Discrete Log caused great excitement in both the physics and computer science communities, many efforts have been put into designing quantum algorithms with exponential speedup over their classical counterparts. However, the progress of this line of research has been disappointingly slower than what people had expected. There are a few quantum algorithms with exponential speedup, such as Hallgren's polynomial-time quantum algorithm for solving Pell's equation (Hallgren 2007) and Kuperberg's ${2}^{O(\sqrt{n})}$ time quantum algorithm for solving the Hidden Subgroup Problem (Lomont 2004) for a dihedral group (Kuperberg 2005); see a recent survey (Childs and van Dam 2010) for a more comprehensive review. But all the problems are number-theoretical or algebraic. (One exception is the oracle separation for the Glued Tree problem by Childs et al. based on a continuous quantum walk (Childs et al. 2003), but the problem is somewhat artificial.) Why does it seem that fast quantum algorithms are much harder to design and what should one do next? In a survey (Shor 2004), Shor gave a couple of possible reasons, including that one does not have enough intuition and experience in dealing with quantum information, and that there may not be many natural problems in quantum computers that can have speedup. He also suggested to try to first study problems that are solvable on a classical computer, aiming at developing algorithmic designing tools with extensive usefulness. It is hoped that studies of BQP-complete problems can shed light on these questions and deepen our understanding of the power and limitation of quantum computation.

BQP is the computational class containing decision problems that are solvable probabilistically on a polynomial-time quantum computer. Since a central problem in quantum computing is the comparison between the computational power of quantum and classical computation, it is natural to see what extra power a quantum computer gives one. In this regard, let BQP-completeness be defined with the reduction being BPP algorithms, that is, those (classical) probabilistic polynomial-time algorithms. In other words, a problem is BQP-complete if it is in BQP, and all other BQP problems can reduce it by probabilistic polynomial-time classical algorithms. Analogous to the fact that NP-complete problems are the “hardest” problems in NP and thus capture the computational power of efficient nondeterministic computation, BQP-complete problems are the hardest problems in BQP and thus capture the computational power of efficient quantum computation.

Like many other “semantic” complexity classes, BQP is not known to contain complete problems. What people usually study for completeness, in such a scenario, is the class containing the promise problems, that is, those decision problems for which the union of Yes and No input instances is not necessarily the whole set of {0,1} strings. In our quantum case, it is the class PromiseBQP, that is, the collection of promise problems solvable in polynomial time on a quantum computer. There are mainly two tracks of PromiseBQP-complete problems. The first track contains problems on the eigenvalues of a local or sparse Hamiltonian or unitary matrix (Wocjan and Zhang 2006; Janzing and Wocjan 2007; Janzing et al. 2008). In the first work (Wocjan and Zhang 2006) along this line, Wocjan and Zhang considered the local Hamiltonian eigenvalue sampling (LHES) and local unitary phase sampling (LUPS) problems: Given a classical string x ∈ {0, 1}ⁿ and a 2ⁿ × 2ⁿ dimensional local Hamiltonian H = ∑ _j H _j or U = ∏ _j U _j , where each H _j or U _j are operating on a constant number of qubits, one is approximately sampling the eigenvalues of H or U under the distribution 〈x|η _j 〉, where |η _j 〉 are the corresponding eigenvectors. These two problems have close connections to other well-known problems in quantum algorithm and complexity theory: LHES is the natural sampling variant of estimating the minimum eigenvalue of a local Hamiltonian, a QMA-complete problem (Kitaev et al. 2002; Kempe et al. 2006), and LUPS is the natural sampling variant of estimating the eigenvalue of a unitary on a given eigenvector, a powerful algorithmic tool called phase estimation (Kitaev 1995). Though not defined as a promise problem, it will be shown in this survey that one can easily transform them into promise problems and thus be PromiseBQP-complete.

Later, the work was extended in two ways. In Janzing et al. (2008), Janzing, Wocjan, and Zhang showed that even if one restricts the local Hamiltonians in LHES to translationally invariant ones operating on a one-dimensional qudit chain, the problem is still PromiseBQP-complete. The other extension, done by Janzing and Wocjan (2007), views the Hamiltonian as a ±1 weighted graph and lifts the graph to power k, where k is part of the input. Then estimating a diagonal entry is PromiseBQP-complete.

The second line of research on PromiseBQP-complete problems is about approximating the Jones polynomial and Tutte polynomial (Freedman et al. 2002b, c; Aharonov and Arad 2006; Aharonov et al. 2006, 2007a; Wocjan and Yard 2006). The Jones polynomial is an important knot invariant with rich connections to topological quantum field theory, statistical physics (Wu 1992) and DNA recombination (Podtelezhnikov et al. 1999). The main result for the Jones polynomial in the studies of PromiseBQP-completeness is that approximating the Jones polynomial of the plat closure of the braid group at e^2πi/k to within some precision is PromiseBQP-complete. Both the efficient quantum algorithm and the universality for constant k were implicitly given by Freedman et al. (2002b, c), but recent results by Aharonov et al. (2006) and Aharonov and Arad (2006) gave an explicit and simpler algorithm and a hardness proof, which also extend the results to any k bounded by a polynomial of the size of the input braid.

The multivariate Tutte polynomial is more general than the Jones polynomial; the Tutte polynomial has many connections to algebraic graph theory and the Potts model in statistical physics. In Aharonov et al. (2007a), Aharonov, Arad, Eban, and Landau gave an efficient algorithm for additively approximating this general polynomial and also showed that the approximation for some ranges is PromiseBQP-complete.

The rest of the chapter is organized as follows. Section 2 gives precise definitions of BQP and PromiseBQP. In Sects. 3 and 4, the two lines of research on PromiseBQP-complete problems are studied. Section 5 concludes with some questions raised.

2 Preliminaries

A promise problem is a pair (L _Yes, L _No) of nonintersecting subsets of {0, 1}*. A language is the set of Yes instances of a promise (L _Yes, L _No) satisfying L _Yes ∪ L _No = {0, 1}*. For more discussions on promise problems versus languages (and why the former is important and sometimes necessary for complexity theory), the readers are referred to Goldreich's survey (Goldreich 2005).

Definition 1 (PromiseBQP) PromiseBQP is the class of promise problems (L _Yes, L _No) such that there is a uniform family of quantum circuits {U _n } operating on p(n) = poly(n) qubits, with the promise that for any n, applying U _n on |x,0^p(n)−n〉 and measuring the first qubit gives the outcome 1 with probability at least 2/3 for all x ∈ L _Yes, and at most 1/3 for all x ∈ L _No.

Definition 2 (BQP) BQP is the class of languages L _Yes with (L _Yes, {0, 1}*−L _Yes) in PromiseBQP.

Like many other “semantic” classes, BQP is not known to have complete problems. However, if one extends languages to promise problems, then PromiseBQP has canonical complete problems for any model, such as the quantum Turing machine (Bernstein and Vazirani 1997), quantum circuit (Yao 1993), adiabatic quantum computer (Aharonov et al. 2004), and quantum one-way computer (Browne and Briegel 2006). For the quantum circuit model, for example, the canonical complete problem is the following.

Definition 3 The canonical complete problem for PromiseBQP in the circuit model has input (〈U〉, x), where 〈U〉 is the description of a uniform family of quantum circuits {U _n } working on p(n) = poly(n) qubits, with the promise that applying U _n on |x,0^p(n)−n〉 and measuring the first qubit gives outcome 1 with probability at least 2/3 for all x ∈ L _Yes and at most 1/3 for all x ∈ L _No. The problem is to distinguish the two cases.

The fact that this is a complete problem for PromiseBQP is almost by definition, which also makes the problem not so interesting. We hope to have more “natural” problems that can help us understand the class PromiseBQP.

3 Approximate Eigenvalue Sampling

This section provides an overview of the first track of studies on PromiseBQP-complete problems, that is on problems about eigenvalues of the local or sparse Hamiltonians. Section 3.1 starts with the local Hamiltonian eigenvalue sampling (LHES) problem, defined in Wocjan and Zhang (2006), and we show that it is PromiseBQP-complete. The problem together with the proof contains the core ideas and techniques for this line of research.

Two extensions are then shown. First, the result in Janzing et al. (2008) is mentioned: even if the Hamiltonians are restricted to be translationally invariant on a one-dimensional qudit chain (with d = O(1)), the problem is still PromiseBQP-complete. Then Sect. 3.2 shows the result in Janzing and Wocjan (2007) that estimating an entry in a sparse real symmetric matrix to some power k, where k is part of the input, is also PromiseBQP-complete.

The proofs in the three sections have similar ingredients, and an attempt is made to unify the notation and give a consistent treatment.

3.1 Phase Sampling and Local Hamiltonian Eigenvalue Sampling

The problems that one is going to see in this section have close relation to two well-known ones, local Hamiltonian minimum eigenvalue and phase estimation, which will be defined below. The local Hamiltonian minimum eigenvalue (LHME) problem is usually abbreviated as the local Hamiltonian (LH) problem, but here LHME is used to distinguish it from the other related problems, which are defined later.

Definition 4 (Local Hamiltonian minimum eigenvalue (LHME)) A tuple (H, a, b) is given where

• 1. H = ∑ _j H _j is a Hamiltonian operating on n qubits, with j ranging over a set of size polynomial in n, and each H _j operating on a constant number of qubits; it is promised that either λ(H) < a or λ(H) > b, where λ(H) is the minimum eigenvalue of H.

• 2. a and b are two real numbers such that a < b and the gap $b - a = \Omega (1/{\rm poly}(n))$.

The task is to distinguish between the case λ(H) < a and the case λ(H) > b.

Definition 5 (Phase estimation (PE)) A unitary matrix U is given by black-boxes of controlled-U, controlled-${U}^{{2}^{2}}, \ldots$, controlled-${U}^{{2}^{t-1} }$ operations, and an eigenvector |u〉 of U with eigenvalue e^2πiφ with the value of φ ∈ [0, 1) unknown. The task is to output an n-bit estimation of φ.

These two problems are both well studied in quantum computing. The Local Hamiltonian problem was shown by Kitaev et al. (2002) to be PromiseQMA-complete when each H _j operates on five qubits; actually it remains PromiseQMA-complete even when each H _j operates on only two qubits (Kempe et al. 2006). Kitaev's efficient quantum algorithm for phase estimation (Kitaev 1995) was a powerful tool for quantum algorithm design, such as for factoring (Shor 1997; Nielsen and Chuang 2000) and some recent quantum-walk based algorithms such as the general quantum walk search (Magniez 2007), formula evaluation (Ambainis et al. 2007), and its extension to span-program evaluation (Reichardt and Spalek 2008).

Let us now consider the sampling variant of the above two problems.

Definition 6 A probability distribution q on ℝ is said to approximate another probability distribution p on a discrete set S ⊆ ℝ with error δ and precision ε if

$${\bf Pr}_{x\leftarrow q}[s - \varepsilon \leq x \leq s + \varepsilon ] \geq (1 - \delta )p(s)$$

(1)

for any s ∈ S.

Intuitively, to approximate the probability distribution p, one draws a sample from another distribution q, and the outcome x is ε-close to s at least (1 −δ) times the correct probability p(s), for each s ∈ S. Now one can define the sampling version of the two problems.

Definition 7 (Local Hamiltonian eigenvalue sampling (LHES)) We are given (H, ε, δ, b) where

• 1. H = ∑ _j H _j is a Hamiltonian operating on n qubits, with j ranging over a set of size polynomial in n, and each H _j operating on a constant number of qubits.

• 2. $\varepsilon = \Omega (1/{\rm poly}(n))$ is the required estimation precision.

• 3. $\delta = \Omega (1/{\rm poly}(n))$ is the required sampling error probability.

• 4. b ∈ {0, 1}ⁿ is a classical n-bit string.

Suppose the eigenvalues and the corresponding eigenvectors of H are {(λ _k , |η _k 〉): k ∈ [2ⁿ]} satisfying |λ _k |< poly(n) for each k. Define the probability distribution D(H, b) over the spectrum of H by

$$D(H,b) =\{ (\lambda, {\bf Pr}(\lambda )): {\bf Pr}(\lambda ) = \sum\limits_{{\lambda }_{k}=\lambda }\vert \langle b\vert {\eta }_{k}\rangle {\vert }^{2}\}.$$

(2)

The task is to draw a sample from some probability distribution p approximating D(H, b) with error δ and precision ε.

Definition 8 (Local unitary phase sampling (LUPS)) We are given (H, ε, δ, b) where

• 1. U is (the description of) an n-qubit quantum circuit.

• 2. $\varepsilon = \Omega (1/{\rm poly}(n))$ is the required estimation precision.

• 3. $\delta = \Omega (1/{\rm poly}(n))$ is the required sampling error probability.

• 4. b ∈ {0, 1}ⁿ is a classical n-bit string.

Suppose the eigenvalues of U are $\{{\lambda }_{j} = {{\rm e}^{2\pi i{\varphi }_{j}}\}}_{j=1,\ldots,{2}^{n }}$ (where φ _j ∈ [0, 1) for each j), with the corresponding eigenvectors $\{\vert {\eta_{j}\rangle \}}_{j=1,\ldots,{2}^{n}}$. The task is to estimate φ _j with error δ and precision ε.

One can immediately see that these two problems are not even promise problems, since the output is a (sampling from a) distribution rather than a Yes/No answer. Nevertheless, they capture the power of efficient quantum computers in the following sense: First, a polynomial-time uniform family of quantum circuits can achieve the sampling requirement. Second, given an oracle for either problem, any BQP problem can be solved by a classical polynomial-time algorithm. These facts will be proved in the following two sections. After that, a mention will be made about how to change LHES to a promise problem so that it really becomes PromiseBQP-complete.

3.1.1 BQP Algorithm for LHES and LUES

In this section we prove that LHES and LUES are solvable by an efficient quantum circuit. The standard algorithm for phase estimation is first reviewed, then it is observed that the same algorithm actually gives the desired LUES solution. This is then used to show an algorithm for LHES.

Phase estimation can be solved by a quantum algorithm as follows. The working space has two registers. The first register consists of $t = n + \lceil \log (2 + 1/2\delta )\rceil$ qubits and is prepared in |0…0〉. The second register contains the eigenvector |u〉. Measuring $\tilde{\varphi }$ in the first register after carrying out the transformations described below gives the desired n-bit estimation of φ with probability of at least 1 −δ.

$$\vert {0\rangle }^{\otimes t}\vert u\rangle$$

(3)

$$\rightarrow {{1} \over {\sqrt{2^{t}}}} \sum\limits_{j=0}^{{2}^{t}-1 }\vert j\rangle \vert u\rangle \quad \quad(\hbox{apply the Fourier transform})$$

(4)

$$\rightarrow {{1} \over {\sqrt{2^{t}}}} \sum\limits_{j=0}^{{2}^{t}-1 }\vert j\rangle {U}^{j}\vert u\rangle \quad \qquad (\hbox{apply the controlled powers of} U)$$

(5)

$$ = {{1} \over {\sqrt{2^{t}}}} \sum\limits_{j=0}^{{2}^{t}-1 }{\rm e}^{2\pi ij\varphi }\vert j\rangle \vert u\rangle$$

(6)

$$\rightarrow \vert \tilde{\varphi }\rangle \vert u\rangle \quad \quad (\hbox{apply the inverse Fourier transform})$$

(7)

The following observation says that the same algorithm actually works for LUES.

Fact If one feeds |0〉|b〉 instead of |0〉|u〉 as input to the above algorithm for the phase estimation problem and $t = \left \lceil \log {1 \over {\varepsilon}} \right\rceil + \left\lceil \log \left(2 + {1 \over {2\delta}} \right) \right\rceil$, then the measurement of the first register gives the desired sampling output. This implies that LUES can be solved by a polynomial-size quantum circuit.

This actually holds not only for |b〉 but also for a general state |η〉. To see why this is true, write |η〉 as $\sum_{j = 1}^n {\alpha _j } | {\eta _j } \left| \right\rangle$〉, then by the linearity of the operations, the final state is ${\alpha }_{j}\vert \tilde{{\varphi }_{j}}\rangle \vert {\eta }_{j}\rangle$. For more details, the readers are referred elsewhere (Nielsen and Chuang 2000; Chap. 5). Note that to implement the controlled-${U}^{{2}^{j}}$ operations for $j = 0,\ldots,{2}^{t} - 1$ in the above algorithm, one needs to run U for 2^t times, which can be done efficiently since $t = \left \lceil \log {1 \over {\varepsilon}} \right \rceil + \left \lceil \log \left( 2 + {1 \over {2\delta}} \right) \right \rceil$ and $\varepsilon = \Omega (1/{\rm poly}(n))$, $\delta = \Omega (1/{\rm poly}(n))$.

Now we can give the efficient algorithm for LHES.

Theorem 1 LHES can be implemented by a uniform family of quantum circuits of polynomial size.

Proof By a simple scaling ($H^\prime = H/\Lambda = \sum\nolimits_{j}{H}_{j}/\Lambda$ where $\Lambda = \max\nolimits_{k}\vert {\lambda }_{k}\vert = {\rm poly}(n)$), one can assume that all the eigenvalues λ _k of H satisfy |λ _k |< 1/4. The basic idea to design the quantum algorithm is to use LUPS on e^2πiH. Note that e^2πiH is unitary, and if the eigenvalues and eigenvectors of H are {λ _k ,|η _k 〉}, then those of e^2πiH are just {e${^{2\pi i\lambda_k}}$, |η _k 〉}. Therefore, it seems that it is enough to run the LUES algorithm on (e^2πiH, ε, δ, b), and if one gets some λ > 1/2, then output λ −1. However, note that H is of exponential dimension, so e^2πiH is not ready to compute in the straightforward way. Fortunately, this issue is well studied in the quantum simulation algorithms, and the standard approach is the following asymptotic approximation using the Trotter formula (Trotter 1959; Chernoff 1968; Nielsen and Chuang 2000) or its variants. Here using the simulation technique, one obtains

$${ \left ({\rm e}^{2\pi i\sum\nolimits_{j}{H}_{j}/m}\right )}^{m} ={ \left (\prod\limits_{j}{\rm e}^{2\pi i{H}_{j}/m}\right )}^{\!\!m} + O(1/m)$$

(8)

Now one runs LUES on (b, ε, δ/2, e^2πiH). Whenever one needs to call e^2πiH, one uses ${\prod }_{j}{\rm e}^{2\pi i{H}_{j}/m}$ for m times instead. Note that such substitution yields O(1/m) deviation, so $t =\log {2 \over {\varepsilon \delta }} + O(1)$ calls yield $O\left ( {1 \over {m\varepsilon \delta }} \right ) \leq {c \over {m\varepsilon \delta}}$ deviation for some constant c. Let $m = {{2c} \over {\varepsilon {\delta}^{2}}}$, thus the final error probability is less than ${\delta \over 2} + {c \over {m\varepsilon \delta}} = \delta$, achieving the desired estimation and sampling precisions.

From the proof one can see that as long as e^2πiH can be simulated efficiently from the description of H, one can sample the eigenvalues of H as desired. Since sparse Hamiltonians, which contain local Hamiltonians as special cases, can be simulated efficiently (Aharonov and Ta-Shma 2003; Berry et al. 2007), we know that if we modify the definition of LHES by allowing H to be sparse then it is also BQP-complete.

3.1.2 BQP Hardness of LHES

Theorem 2 PromiseBQP ⊆ P ^LHES

Proof For any L ∈ PromiseBQP, there is a uniform family of polynomial size quantum circuits with ε-bounded error (for a small constant ε) that decides if x ∈ L or x ∉ L. Denote by U the corresponding quantum circuit (of size n) and suppose the size of U is M, which is bounded by a polynomial in n. Further suppose that the computation is described by $U\vert x,{\bf 0}\rangle = {\alpha }_{x,0}\vert 0\rangle \vert {\psi }_{x,0}\rangle + {\alpha }_{x,1}\vert 1\rangle \vert {\psi }_{x,1}\rangle$, where 0 is the initial state of the ancillary qubits, and |ψ _x,0〉 and |ψ _x,1〉 are pure states. After the U transform, the first qubit is measured and the algorithm outputs the result based on the outcome of the measurement. The correctness of the algorithm requires that |α _x,0|² < ε if x ∈ L, and |α _x,1|² < ε if x ∉ L.

Now, a local Hamiltonian H is constructed encoding the circuit U and a binary string b encoding the inputs x, such that eigenvalue sampling applied to H and b yields significantly different probability distributions for the two cases of x ∈ L and x ∉ L.

To this end, the circuit V is constructed as follows: one first applies the original circuit U, then flips the sign for the component with the first qubit being 1, and finally reverses the computation of U. See Fig. 1 .

Fig. 1

Circuit V.

Suppose the original circuit U is decomposed as the product of elementary gates, that is, $U = {U}_{\!m-1}\ldots {U}_{0}$, where each U _j is an elementary gate. Let V _j be the j-th gate in V, where j = 0, 1, …, 2m. Let $M = 2m + 1$, the number of gates in V. Attach a clock register to the system and define the operator

$$F = \sum\limits_{j\ =\ 0}^{M-1}{V }_{ j} \otimes \vert j + 1\rangle \langle j\vert$$

(9)

where the summation in the index is module M. Note that F is an O(logN)-local operator; we will remark how to slightly modify it to be a 4-local operator at the end of the proof. Define

$$\vert {\varphi }_{x,j}\rangle = {F}^{j}(\vert x, {\bf 0}\rangle \vert 0\rangle )$$

(10)

for j ≥ 0, where F ^j means that F is applied j times. It is easy to see that |φ _{x, j} 〉 = |φ _{x, j {mod}2M} 〉 for j ≥ 2M.

Note that due to the clock register for different j and j ′ in {0,..., 2M −1}, |φ _{x, j} 〉 and |φ _{x,j ′} 〉 are orthogonal if |j−j ′| ≠ M. Also note that ${U}_{\!m-1}...{U}_{0}\vert x,{\bf 0}\rangle = U\vert x,{\bf 0}\rangle =$ ${\alpha }_{x,0}\vert 0\rangle \vert {\psi }_{x,0}\rangle + {\alpha }_{x,1}\vert 1\rangle \vert {\psi }_{x,1}\rangle$ by definition. Therefore, we have

$$\langle {\varphi }_{x,j}\vert {\varphi }_{x,M+j}\rangle =\langle x,{\bf 0}\vert {U}_{0}^{\dagger }...{U}_{ m-1}^{\dagger }({P}_{ 0} - {P}_{1}){U}_{\!m-1}...{U}_{0}\vert x, {\bf 0}\rangle = \vert {\alpha }_{x,0}{\vert }^{2} -\vert {\alpha }_{ x,1}{\vert }^{2}$$

(11)

for $j = 0,...,M - 1$, where P ₀ and P ₁ are the projections onto the subspaces corresponding to the first qubit being 0 and 1, respectively. To summarize, we have

$$\langle {\varphi }_{x,j}\vert {\varphi }_{x,j^{\prime}}\rangle =\left\{\begin{array}{ll}0 & \vert j - j^{\prime}\vert \neq M \cr \vert {\alpha}_{x,0}{\vert }^{2} -\vert {\alpha }_{x,1}{\vert }^{2}& \vert j - j^{\prime}\vert = M \end{array}\right.$$

(12)

Define the subspace ${S}_{x} = {\rm span}\{\vert {\varphi }_{x,j}\rangle: j = 0,\ldots,2M - 1\}$. The key property here is that though F has an exponentially large dimension, $F{\vert }_{{S}_{x}}$ is of only polynomial dimension. Depending on the probability, we consider three cases.

Case |α _x,0|= 1: The dimension of S _x is M, and $F{\vert }_{{S}_{x}}$ is a shift operator on the basis $\{\vert {\varphi {}_{x,j}\rangle \}}_{j=0,...,M-1}$, that is $F\vert {\varphi }_{x,j}\rangle = \vert {\varphi }_{x,j+1 \ {\rm mod} \ M}\rangle$. It is not hard to see that this operator has eigenvalues and the corresponding eigenvectors

$${\lambda }_{k} = {\omega }_{M}^{k},\quad \vert {\xi }_{ k}\rangle = {1 \over {\sqrt{M}}} \sum\limits_{j\ =\ 0}^{M-1}{\omega }_{ M}^{-kj}\vert {\varphi }_{ x,j}\rangle,\quad k = 0,1,\ldots, M - 1$$

(13)

where ${\omega }_{M} = {\rm e}^{2\pi i/M}$.

Case α _x,0 = 0: The dimension of S _x is M, and $F{\vert }_{{S}_{x}}$ is almost a shift operator on the basis $\{\vert {\varphi_{x,j}\rangle \}}_{j=0,...,M-1}$: $F\vert {\varphi }_{x,j}\rangle = \vert {\varphi }_{x,j+1}\rangle$ for all $j = 0,\ldots,M - 2$, and $F\vert {\varphi }_{x,M-1}\rangle = -\vert {\varphi }_{x,0}\rangle$. It is not hard to see that this operator has eigenvalues and the corresponding eigenvectors

$${\mu }_{k} = {\omega }_{M}^{k+1/2},\quad \vert {\eta }_{ k}\rangle = {1 \over {\sqrt{M}}} \sum\limits_{j=0}^{M-1}{\omega }_{ M}^{-(k+1/2)j}\vert {\varphi }_{ x,j}\rangle,\quad k = 0,1,\ldots, M - 1.$$

(14)

Case 0 < |α _x,0 |< 1: In this general case, S _x has dimension 2M. To find the eigenvalues and eigenvectors of $F{\vert }_{{S}_{x}}$, define

$$\vert {\phi }_{x,j}\rangle = {{\vert {\varphi }_{x,j}\rangle + \vert {\varphi }_{x,M+j}\rangle } \over {\|\vert {\varphi }_{x,j}\rangle + \vert {\varphi }_{x,M+j}\rangle \|}},\quad \vert {\gamma }_{x,j}\rangle = {{\vert {\varphi }_{x,j}\rangle -\vert {\varphi }_{x,M+j}\rangle } \over {\|\vert {\varphi }_{x,j}\rangle -\vert {\varphi }_{x,M+j}\rangle \|}}$$

(15)

for $j = 0,...,M - 1$. Then first, because $\langle {\varphi }_{x,j}\vert {\varphi }_{x,N+j}\rangle = \vert {\alpha }_{x,0}{\vert }^{2} -\vert {\alpha }_{x,1}{\vert }^{2}$ is a real number, we have 〈φ _x,j |γ _x,j 〉 = 0. Together with Eq. 12, we know that

$$\{\vert {\phi }_{x,0}\rangle,\ldots,\vert {\phi }_{x,M-1}\rangle,\vert {\gamma }_{x,0}\rangle,\ldots,\vert {\gamma }_{x,M-1}\rangle \}$$

(16)

forms an orthonormal basis of S _x . One can further observe that $F{\vert }_{{S}_{x}} = {F}_{0} \oplus {F}_{1}$, where F ₀ and F ₁ act on ${S}_{x,+} = {\rm span}\{\vert {\phi }_{x,0}\rangle,...,\vert {\phi }_{x,M-1}\rangle \}$ and ${S}_{x,-} = {\rm span}\{\vert {\gamma }_{x,0}\rangle,...,\vert {\gamma }_{x,M-1}\rangle \}$, respectively, with the matrix representations (in the basis {|φ _x,j 〉} and {|γ _x,j 〉}, respectively) as follows:

$${ F}_{0} = \left ( \matrix {0 & 1 & 0 & \ldots & 0 & 0 \cr 0 & 0 & 1 & \ldots & 0 & 0 \cr \vdots & \vdots & \vdots && \vdots & \vdots \cr 0 & 0 & 0 & \ldots & 0 & 1 \cr 1 & 0 & 0 & \ldots & 0 & 0} \right ), \qquad {F}_{1} = \left (\matrix {0 & 1 & 0 & \ldots & 0 & 0 \cr 0 & 0 & 1 & \ldots & 0 & 0 \cr \vdots & \vdots & \vdots & & \vdots & \vdots \cr 0 & 0 & 0 & \ldots & 0 & 1 \cr -1 & 0 & 0 & \ldots & 0 & 0} \right)$$

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So F ₀ and F ₁ have the same matrix representations as the two operators in the previous two cases, thus having the same eigenvalues and eigenvectors (with respect to different basis vectors though). Precisely, F ₀ has eigenvalues λ _k with eigenvectors $\vert {\xi }_{k}\rangle = {1 \over {\sqrt{M}}}\sum\nolimits_{j=0}^{M-1}{\omega }_{M}^{-kj}\vert {\phi }_{x,j}\rangle$; F ₁ has eigenvalues μ _k with eigenvectors $\vert {\eta }_{k}\rangle = {1 \over {\sqrt{M}}}\sum\nolimits_{j=0}^{M-1}{\omega }_{M}^{-(k+1/2)j}\vert {\gamma}_{x,j}\rangle$. (Here one uses the same notation |ξ _k 〉 and |η _k 〉 because they are consistent with the previous cases when |α _x,0|² = 0 or 1 respectively.)

By this, it is not hard to find the eigenvalue sampling probabilities:

$$c\vert \langle {\varphi }_{x,0}\vert {\xi }_{k}\rangle {\vert }^{2} = \vert \langle {\varphi }_{ x,0}\vert {1 \over M}\sum\limits_{j}{w}_{M}^{-kj}{\phi }_{ x,j}\rangle {\vert }^{2}$$

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$$= {1 \over M}\vert \langle {\varphi }_{x,0}\vert {\phi }_{x,0}\rangle {\vert }^{2} \qquad ({\rm by \ Eq.} \ 12)$$

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$$\hskip11.5pc= {1 \over M} \left\vert \left\langle{\varphi }_{x,0} \left\vert {{{\varphi }_{x,0} +{\varphi }_{x,M}} \over {\|\vert {\varphi }_{x,0}\rangle + \vert{\varphi }_{x,M}\rangle \|}} \right. \right\rangle \right\vert^{2}\qquad {({\rm by \ def \ of} \ \vert {\phi }_{ x,0}\rangle)}$$

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$$= {{\vert {\alpha }_{x,0}{\vert }^{2}} \over {M}} \qquad {({\rm by \ Eq}. \ 12)}$$

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and similarly we have

$$\vert \langle {\varphi }_{x,0}\vert {\eta }_{k}\rangle {\vert }^{2} = {{\vert {\alpha }_{x,1}{\vert }^{2}} \over {M}}$$

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Now the local Hamiltonian is constructed as

$$H = (F + {F}^{\dagger })/2.$$

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It is easy to verify that H is a local Hamiltonian. And further, suppose the eigenvalues of F are {κ _j } with the eigenvectors {|ψ _j 〉}, then the eigenvalues of H are just $\{({\kappa }_{j} + {\kappa }_{j}^{{_\ast}})/2\}$ with the same corresponding eigenvectors. Thus H has eigenvalues cos(2kπ/M) and $\cos ((2k + 1)\pi /M)$ for $k = 0,1,\ldots \,,(M - 1)/2$, and the distribution D(H, |φ _x,0〉) is {cos(2kπ/M) with probability |α _x,0|²/M, and $\cos ((2k + 1)\pi /M)$ with probability |α _x,1|²/M}.

Now if x ∈ L _Yes, then |α _x,0|² is very small, thus with high probability the sample gives a random value uniformly chosen from $\{\cos (2k\pi /M)\!\!: k = 0,\ldots \,,(M - 1)/2\}$. If x ∈ L _No, then |α _x,1|² is very small, thus with high probability the sample gives a random value uniformly chosen from $\{\cos ((2k + 1)\pi /M)\!\!: k = 0,\ldots \,,(M - 1)/2\}$. Since any two values chosen from these two sets are at least Ω(M ⁻²) away from each other, an approximator with precision O(M ⁻²) suffices to distinguish between these two cases.

Finally, to obtain a 4-local LHES, |i〉 is replaced by |e _i 〉 = |0 … 010 … 0〉 for $i = 0,\ldots,N - 1$, where the only 1 appears at coordinate i. Modify the operator F to be

$$F = \sum\limits_{j=0}^{M-1}{V }_{ j} \otimes \vert {e}_{j+1}\rangle \langle {e}_{j}\vert $$

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Note that |e _j 〉〈e _j−1| and |e ₀〉〈e _N−1| are 2-local. The remaining proof passes through. This completes the proof for Theorem 2.

3.1.2.1 Changing It to a Promise Problem

Now that we have completed both the algorithm and the completeness, it is not hard to see that the sampling problem can be modified to a promise problem as follows. We can derive its PromiseBQP-completeness from the above proofs. Since this was not mentioned in the previous work (Wocjan and Zhang 2006), a bit more detail is given below. Let

$$\eqalign { & {p}_{0}(M) = {\rm uniform \ distribution \ over} \ \{ \cos (2k\pi /M)\!\!: k = 0,1,\ldots \,,(M - 1)/2\}; \cr & {p}_{1}(M) = {\rm uniform \ distribution \ over} \ \{ \cos ((2k + 1)\pi /M)\!\!: k = 0,1,\ldots, (M - 1)/2\}.}$$

Definition 9 (Local Hamiltonian eigenvalue distribution (LHED)). We are given (H, ε, δ, b, M) where

• 1. H = ∑ _j H _j is a Hamiltonian operating on n qubits, with j ranging over a set of size polynomial in n, and each H _j operating on a constant number of qubits.

• 2. $\varepsilon = \Omega (1/{\rm poly}(n))$, $\varepsilon = o({M}^{-2})$ is the required estimation precision.

• 3. $\delta = \Omega (1/{\rm poly}(n))$, δ < 1/10 is the required sampling error probability.

• 4. b ∈ {0, 1}ⁿ is a classical n-bit string.

• 5. M = poly (n).

Suppose the eigenvalues and the corresponding eigenvectors of H are {(λ _k ,|η _k 〉): k ∈ [2ⁿ]} satisfying |λ _k |< poly (n) for each k. The input has the promise that the probability distribution D(H, b), given by Eq. 2, approximates exactly one of p ₀(M) and p ₁(M) with error δ and precision ε. The task is to determine which one, p ₀(M) or p ₁(M), D(H, b) approximates.

3.1.2.2 Restricted to One-Dimension Chain

It turns out that even if one restricts the LHES problem to the Hamiltonian on a one-dimensional qudit chain for d = O(1), the problem is still PromiseBQP-complete (Janzing et al. 2003). The Hamiltonian is constructed along the same lines as above, with some additional treatment, following the ideas of (Aharonov et al. 2007b), to make it translationally invariant on a one-dimensional qudit chain.

3.2 Lifting the Matrix to a Power

One may find the sampling nature of the previous problems not natural, and would like to consider the average eigenvalue of a local Hamiltonian. (The authors of Wocjan and Zhang (2006) gave the credit for the question about the average eigenvalue to Yaoyun Shi.) If the distribution is, as before, induced by a vector |b〉, such that the average is $\sum\nolimits_k {|\left\langle {b|\left. {\xi _k } \right\rangle } \right.} |^2 \lambda _k$, then it is not an interesting problem to study for PromiseBQP: Note that the value is equal to 〈b|H|b〉, and further

$$\langle b\vert H\vert b\rangle =\langle b\vert \sum\limits_{j}{H}_{j}\vert b\rangle = \sum\limits_{j}\langle b\vert {H}_{j}\vert b\rangle$$

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Since each 〈b|H _j |b〉 can be easily computed even deterministically, one can obtain the exact average eigenvalue of a local Hamiltonian deterministically in polynomial time.

However, if one raises the matrix to some power, then the problem becomes PromiseBQP-complete again. This is the subject of this section, based on the result in Janzing and Wocjan (2007).

Definition 10 The sparse matrix powered entry (SMPE) problem is defined as follows. The input is a tuple (A, b, m, j, ε, g) where

• 1. A ∈ ℝ^N×N is a symmetric matrix with the operator norm $\|A\| \leq b$, A has no more than s = polylog(N) nonzero entries in each row, and there is an efficiently computable function f specifying for each given row the nonzero entries and their positions.

• 2. m = polylog(N) is a positive integer, j ∈ [N], $\varepsilon = 1/{\rm polylog}(N)$ and g ∈ [−b ^m, b ^m].

The input has the promise that either $\left( {A^m } \right)_{jj} \ge g + \varepsilon b^m$ or $\left( {A^m } \right)_{jj}\ \le\ g - \varepsilon b^m$. The task is to distinguish between these two cases.

The main theorem is now given.

Theorem 3 The problem SMPE is PromiseBQP-complete.

The algorithm is very similar to the one in the last section, except that now the efficient quantum algorithm is used to simulate the process e ^−iAt for general sparse Hamiltonians A (Berry et al. 2007). Next, an account of the proof of the PromiseBQP-hardness is given.

From the definition of H in Eq. 23, one can see that

$${H}^{t} = \sum\limits_{k}\cos^{t}(2k\pi /M)\vert {\xi}_{k} \rangle \langle {\xi}_{k}\vert + \sum\limits_{k}\cos^{t}((2k + 1)\pi /M)\vert {\eta}_{k}\rangle \langle {\eta}_{k}\vert$$

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Thus the distribution D(H ^t,|φ _x,0〉) is {cos^t(2kπ/M) with probability |α _x,0|²/M, and ${\cos }^{t}((2k + 1)\pi /M)$ with probability |α _x,1|²/M}, whose average is

$$\eqalign { \langle {\varphi }_{x,0}\vert {H}^{t}\vert {\varphi }_{ x,0}\rangle = & {1 \over M}\vert {\alpha }_{x,0}{\vert }^{2}\left(1 + \sum\limits_{k=1}^{(M-1)/2} \cos^{t}(2k\pi /M)\right) \cr & \quad + {1 \over M}\vert {\alpha }_{x,1}{\vert }^{2}\left( - 1 + \sum\limits_{k=0}^{(M-3)/2} \cos^{t}((2k + 1)\pi /M)\right)}$$

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When t is large enough, say t = M ³, all the cosine terms combined become negligible compared to the 1 or −1 in the summation. Thus, depending on whether |α _x,1|² is close to 1 or 0, the average eigenvalue $\left\langle {\varphi _{x,0} |H^t |\varphi _{x,0} } \right\rangle$ will be close to either $-\!1/M$ or 1/M, respectively. Therefore, estimating the average eigenvalue can determine whether |α _x,1|² is close to 1 or 0, solving the starting PromiseBQP problem.

4 Additive Approximation of the Jones Polynomials and Tutte Polynomials

A completely different vein of research on PromiseBQP-completeness is the study of approximation of Jones polynomials and Tutte polynomials (Freedman 2002b, c; Aharonov et al. 2006, 2007a; Aharonov and Arad 2006; Wocjan and Yard 2008). There are at least two interesting aspects of this line compared to the problems discussed in the previous section. First, the problems look less quantum, at least by the definition. Second, the algorithms do not use quantum Fourier transform as many other quantum algorithms with exponential speedup over their classical counterparts do. What the algorithms use is the homomorphism property of a representation.

Next, this line of research is introduced in Sect. 4.1 to give some background about the Jones polynomial and Tutte polynomial, mainly about their connections to physics and combinatorics. Section 4.2 presents the efficient quantum algorithms for approximating Jones polynomials of trace closure of the braid group at some roots of unity. Finally a brief account is given of two subsequent (unpublished) results: the PromiseBQP-hardness of approximating Jones polynomials (Aharonov and Arad 2006) in Sect. 4.3 and the algorithms and complexity of approximating the Tutte polynomials (Aharonov et al. 2007a) in Sect. 4.4.

4.1 About Jones Polynomials and Tutte Polynomials

A short discussion about the connections of the Jones polynomial and the Tutte polynomial to other fields is given. The definitions are a bit involved and thus deferred to later sections.

In knot theory, the Jones polynomial is a knot invariant discovered by Jones (1985). Specifically, it is, for each oriented knot or link, a Laurent polynomial in the variable $\sqrt{t}$ with integer coefficients. The Jones polynomial is an important knot invariant in low dimensional topology; it is also related to statistical physics (Wu 1992) and DNA recombination (Podtelezhnikov et al. 1999).

The connection between the Jones polynomial and quantum computing is already known. Freedman et al. (2002b, c) showed that a model of quantum computing based on topological quantum field theory and Chern–Simons theory (Freedman 1998; Freedman et al. 2002a) is equivalent to the standard quantum computation model up to a polynomial relation. Note that these results actually already imply an efficient quantum algorithm for approximating the Jones polynomial at e^2πi/5, though the algorithm was not explicitly given. In Aharonov et al. (2006b), Aharonov, Jones and Landau gave a simple and explicit quantum algorithm to approximate the Jones polynomial at all points of the form e^2πi/k, even if k grows polynomially with n. At the universality side, the result in Freedman et al. (2002) implies that approximating the Jones polynomial of the plat closure of a braid at e^2πi/k is PromiseBQP-hard for any constant k. In Aharonov and Arad (2006), Aharonov and Arad generalized this by showing that the problem is PromiseBQP-hard also for asymptotically growing k, bounded by a polynomial of size of the braid. These together give a new class of PromiseBQP-complete problems.

The usual Tutte polynomial is a two-variable polynomial defined for graphs (or more generally for matroids). It is, essentially, the ordinary generating function for the number of edge sets of a given size and connected components. Containing information about how the graph is connected, it plays an important role in algebraic graph theory. It contains as special cases several other famous polynomials, such as the chromatic polynomial, the flow polynomial and the reliability polynomial, and it is equivalent to Whitney's rank polynomial, Tutte's own dichromatic polynomial, and the Fortuin–Kasteleyn's random-cluster model under simple transformations. It is the most general graph invariant defined by a deletion–contraction recurrence. See textbooks (Bollobás 1998; Biggs 1993; Godsil and Royle 2001) for more detailed treatment.

The result in Aharonov et al. (2007a) focuses on the multivariate Tutte polynomial, which generalizes the two-variable case by assigning to each edge a different variable; see survey (Sokal 2005). This generalized version has rich connections to the Potts model in statistical physics (Wu 1982).

On the complexity side, the exact evaluation of the two-variable Tutte polynomial for planar graphs is #P-hard. Both positive and negative results are known for the multiplicative approximation FPRAS (fully polynomial randomized approximation scheme); see Aharonov et al. (2007a) for more details.

4.2 Additive Approximation of the Jones Polynomials

4.2.1 Definitions and the Main Theorems

This section presents an efficient quantum algorithm to additively approximate the Jones polynomial of trace and plat closures of braids at roots of unity. We begin by defining the Jones polynomial. A set of circles embedded in $\mathbb R^3$ is called a link L. If each circle has a direction, then we say that the link is oriented. A link invariant is a function on links that is invariant under isotopy of links. That is, the function remains unchanged when the links are distorted (without being broken). An important link invariant is the Jones polynomial V _L(t), which is a Laurent polynomial in $\sqrt{t}$ over $\mathbb Z$.

Definition 11 (Writhe) The writhe w(L) of an oriented link L is the number of positive crossings (l ₊ in Fig. 2 ) minus the number of negative crossings (l ₋ in Fig. 2 ) of L.

Fig. 2

Positive and negative crossings.

Definition 12 (Bracket Kauffman polynomial) The bracket Kauffman polynomial 〈L〉 of an oriented link L is

$$\langle L\rangle =\sum \limits_{\sigma }\sigma (L)$$

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where the summation is over all the crossing-breaking states. A crossing-breaking state σ is an n-bit string indicating how to break all crossings by changing each crossing in one of the two ways as shown in Fig. 3 . Define $\sigma (L) = {A}^{{\sigma }_{+}-{\sigma }_{-}}{d}^{\vert \sigma \vert -1}$, where σ ₊ and σ ₋ are the numbers of choices of the first and the second case respectively, when breaking all crossings in L, |σ|is the number of closed loops in the resulting link, and $d = -{A}^{2} - {A}^{-2}$.

Fig. 3

Two ways to break a crossing.

Definition 13 (Jones polynomial) The Jones polynomial of an oriented link L is

$${V }_{\rm L}(t) = {V }_{\rm L}({A}^{-4}) = {(-A)}^{3w(L)}\langle L\rangle$$

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where w(L) is the writhe of L, and 〈L〉 is the bracket Kauffman polynomial of L.

There are different ways of forming links. In particular, one can obtain a link from a braid. First, an intuitive geometrical definition of a braid is given here. Consider two horizontal bars each with n pegs, one on top of the other; see Fig. 4 . The pegs on the upper bar are called the upper pegs, and we index them from left to right using i = 1, …, n; similarly for the lower pegs. Each strand goes from an upper peg, always downwards, to a lower peg, so that finally each peg has exactly one strand attached to it. The n-strand braid group B _n is the set of n-strand braids with the multiplication defined by putting the first braid on top of the second. To be more precise, for braids b ₁ and b ₂, the new braid b ₁ ⋅ b ₂ is obtained by putting b ₁ on top of b ₂, and removing the pegs of the lower bar of b ₁ and the upper bar of b ₂ with strands on the same peg connected.

Fig. 4

A braid with 4 strands.

The braid group B _n has an algebraic presentation by generators and relations.

Definition 14 Let B _n be the group with generators {1, σ ₁,..., σ _n−1} with relations

• 1. σ _i σ _j = σ _j σ _i for all |i −j |≥ 2

• 2. ${\sigma }_{i}{\sigma }_{i+1}{\sigma }_{i} = {\sigma }_{i+1}{\sigma }_{i}{\sigma }_{i+1}$

The correspondence between the geometrical pictures and the algebraic presentation is as follows: 1 corresponds to the braid where all strands j go from upper peg j to lower peg j without crossing any other strand; σ _i corresponds to the same braid except that strand i and i + 1 have one crossing, with strand i in front. The above two relations are easily verified by this correspondence.

Braids are not links since the strands are not closed circles, but there are different ways of forming links from braids. Two simple ones are the trace closure and the plat closure. The trace closure B ^tr of a braid B connects upper peg i and lower peg i without crossing any strand in b. The plat closure B ^pl of a 2n-strand braid B connects its upper peg 2i −1 and upper peg 2i for all i ∈ [n]; similarly for the lower pegs. See Fig. 5 .

Fig. 5

The trace and plat closures of a 4-braid.

The main theorem of this section says that there are quantum algorithms additively approximating the Jones polynomials of B ^tr and B ^pl at ${A}^{-4} = {\rm e}^{2\pi i/k}$ within some precision.

Theorem 4 There is a quantum algorithm, which on input n-strand m-crossing braid B outputs a complex number c satisfying $\vert c - {V }_{{\!\!B}^{{\rm tr}}}({\rm e}^{2\pi i/k})\vert < \varepsilon {(-{A}^{2} - {A}^{-2})}^{n-1}$ with probability $1 - 1/exp(n,m,k)$ for some $\varepsilon = 1/{\rm poly}(n,m,k)$.

Theorem 5 There is a quantum algorithm, which on input n-strand m-crossing braid B outputs a complex number c satisfying $\vert c - {V }_{{B}^{{\rm pl}}}({\rm e}^{2\pi i/k})\vert < \varepsilon {(-{A}^{2} - {A}^{-2})}^{3n/2}/N$ with probability $1 - 1/exp(n,m,k)$ for some $\varepsilon = 1/{\rm poly}(n,m,k)$ . Here

$$N =\sum \limits_{l=1}^{k-1}\sin (\pi l/k)\vert {P}_{ n,k,l}\vert$$

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where

$${ P}_{n,k,l} = \left\{ x \in \{ 0,{1\}}^{n}: 1 \leq 2\sum \limits_{i=1}^{j}{x}_{ i} -j \leq k-1,\forall j \in [n];2\sum \limits_{i=1}^{n}{x}_{ i} -n = l \right\}$$

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4.2.2 The Approximation Algorithms

4.2.2.1 Main Idea and Overview

The main idea of designing the efficient quantum is to design a representation of the braid group and use the homomorphism property of the representation to decompose the computation into ones for each crossing in the braid, which can be done efficiently. This idea was also used to design classical algorithms. Let us recall an example to illustrate the idea. Consider the word problem of the free group generated by {a, b}. That is, given a sequence of elements from {a, b}, decide whether the multiplication (in that order) is the identity element of the group. There is a simple linear time algorithm using the stack (or pushdown machine, in other words), but the algorithm also uses linear space. Interestingly, Lipton and Zalcstein showed in 1977 that it can be solved in log space in a very cute manner. Basically, they used the following theorem proved by Ivan Sanov in 1947.

Theorem 6 There are two integer matrices A, B so that the mapping a → A and b → B is a faithful representation of the free group on {a, b}.

That means that we can replace the word problem by this question: Does a sequence of matrices over $\{A,B,{A}^{-1},{B}^{-1}\}$ equal the identity matrix I ? Since multiplication of 2 × 2 matrices does not need the stack anymore, this basically gives a log space algorithm. (There is actually one more difficulty: during the computation there may be entries that are too large and cannot be stored in log-space. The solution is to do the multiplication mod p for all p using O(log(n)) bits. Then the correctness is guaranteed by the Chinese Remainder Theorem and the following well-known fact: ∏ _p≤t p ≥ c ^t, where the product is over primes and c > 1.)

The quantum algorithm to approximate the Jones polynomial also uses representation of the braid group. A high level picture is:

$${B}_{n}{\mathop {\longrightarrow}^{\rho_{\rm A}}} {\rm TL}_{n}(d){\mathop {\longrightarrow}^{\Phi}} \mathbb C^{r\times r}$$

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Here, TL _n (d ) is the Temperley–Lieb algebra, which will be defined later. The mapping ρ _A is a homomorphism from B _n to TL _n (d ). The mapping Φ is the representation of TL _n (d) by r × r unitary matrices for some r.

The analysis can be divided into three steps:

• 1. For the link B ^tr, the Jones polynomial is

  $${V}_{{\!\!B}^{{\rm tr}}}({A}^{-4}) = {(-A)}^{3w({B}^{{\rm tr}}) }{d}^{n-1}tr({\rho}_{A}(B))$$

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  The function tr(⋅) is some mapping from TL _n (d) to ℂ to be defined later. By this equality, it is enough to approximate tr(ρ _A (B)).

• 2. One can calculate tr(ρ _A (B)) by

  $$tr({\rho}_{A}(B)) = T{r}_{n}(\Phi \circ {\rho }_{A}(B))$$

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  where $\circ$ is the standard function composition, and Tr _n is defined as follows. For a matrix W ∈ ℂ^r×r,

  $$T{r}_{n}(\Phi \circ {\rho }_{A}(B)) = {1 \over N} \sum\limits_{l=1}^{k-1}\sin (\pi l/k)Tr(\Phi \circ {\rho }_{ A}(B){\vert}_{l})$$

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  where Tr is the standard matrix trace, N is as defined in Theorem 5, W | _l is W restricted on some subspace. If an orthonormal basis of Φ ∘ ρ _A (B)| _l is {|p〉}, then one can further the calculation by

  $$Tr(\Phi \circ {\rho }_{A}(B){\vert }_{l}) = \sum\limits_{p}\langle p\vert \Phi \circ {\rho }_{A}(B)\vert p\rangle$$

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  It turns out that for each l, one can efficiently sample p almost uniformly at random (even on a classical computer). One also needs the following well-known fact in quantum computing.

  Fact Given a quantum state |ψ〉 and a quantum circuit Q, one can generate a random variable b ∈ ℂ with |b|≤ 1 and E[b] = 〈ψ|Q|ψ〉.

  By the above analysis, one can achieve the approximation as long as one can implement Φ ∘ ρ _A (B) efficiently on a quantum computer.

• 3. This is where the homomorphism comes into the picture. Each braid B can be decomposed into the product of a sequence of basic braids σ _i . The A and Φ are carefully designed such that Φ ∘ ρ _A is a unitary representation. Thus $\Phi \circ {\rho}_{A}(B) = \Phi \circ {\rho }_{A}\left({\prod }_{i}{\sigma }_{i} \right) ={ \prod }_{i}\Phi \circ {\rho }_{A}({\sigma }_{i})$. Therefore, it is enough to implement Φ ∘ ρ _A (σ _i ) for each basis σ _i , and it turns out that this is not hard to do. This finishes the overview of the whole idea of designing the quantum algorithm.

4.2.2.2 More Details

Next, more details to carry out the above strategy are given. To start with, the Temperley–Lieb algebra is defined.

Definition 15 (Temperley–Lieb algebra) The Temperley–Lieb algebra TL _n (d) is the algebra generated by {1, E ₁,..., E _n−1} with relations

• 1. E _i E _j = E _j E _i , |i −j| ≥ 2

• 2. ${E}_{i}{E}_{i+1}{E}_{i} = {E}_{i}{E}_{i-1}{E}_{i} = {E}_{i}$

• 3. E _i ² = dE _i

As the braid group B _n , the Temperley–Lieb algebra also has a nice pictorial description called Kauffman diagrams. A Kauffman n-diagram has an upper bar with n top pegs and a lower bar with n lower pegs as in a braid, but it does not have crossings or loops. A typical Kauffman 4-diagram is as shown in Fig. 6 .

Fig. 6

A typical Kauffman 4-diagram.

The multiplication of two Kauffman n-diagrams K ₁ ⋅K ₂ is obtained similarly for braids: that is, putting K ₁ on top of K ₂. Note that K ₁ ⋅ K ₂ now may contain loops. If there are m loops, then the loops are removed and we add a factor of d ^m in the resulting diagram. See Fig. 7 . The Kauffman diagrams over ℂ form an algebra. The correspondence between the pictures and the original definition of Temperley–Lieb algebra is shown in Fig. 8 .

Fig. 7

Product of two Kauffman 4-diagrams.

Fig. 8

Correspondence between the pictures and the Temperley–Lieb algebra.

The first mapping ρ _A is now defined in Eq. 32.

Definition 16 ρ _A : B _n ↦ TL _n (d) is defined by ${\rho }_{A}({\sigma }_{i}) = A{E}_{i} + {A}^{-1}1$.

The following facts are easily verified.

Fact 1 The mapping ρ _A respects the two relations in the definition of the braid group.

Fact 2 If |A|= 1 and Φ(E _i ) is Hermitian for each i, then the map Φ ∘ ρ _A is a unitary representation of B _n .

Now the trace function in Step 1 and 2 is defined. It is easier to define the Kauffman diagrams.

Definition 17 The Markov trace tr: TL _n (d) → ℂ on a Kauffman n-diagram K connects the upper n pegs to the lower n pegs of K with nonintersecting curves, as in the trace closure case. Then $tr(K) = {d}^{a-n}$, where a is the number of loops of the resulting diagram. Extend tr to all of TL _n (d) by linearity.

For this definition, it is not hard to check that the equality in Step 1 holds.

Lemma 1 For any braid B, one has

$${V }_{{\!\!B}^{\rm tr}}({A}^{-4}) = {(-A)}^{3w({B}^{{\rm tr}}) }{d}^{n-1}tr({\rho }_{ A}(B))$$

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Now the second mapping Φ is defined, for which the path model representation of TL _n (d ) will be needed. Consider a graph L _k−1 of a line with k −1 points (connected by k −2 edges); see Fig. 9 . Let Q _n,k be the set of all paths of length n on L _k−1. That is, Q _n,k can be identified with $\{q \in {[k - 1]}^{n+1}\!\!: \vert {q}_{i} - {q}_{i+1}\vert = 1,\forall i \in [n - 1]\}$. Let r = |Q _n,k |. That is, the size of Q _n,k is the dimension of matrix Φ(K ); thus one can index the row/column of the matrix by a path in Q _n,k . To specify Φ easily, let one use the bit string representation of a path. Each path is specified by a string p ∈ {0, 1}ⁿ such that p _i = 1 if the i-th step goes right and p _i = 0 if the i-th step goes left. To guarantee that the path is always within the graph, one requires 1 ≤ z _j ≤ k −1 for any j ∈ [n], where ${z}_{j} = 2 \sum\nolimits_{i=1}^{j}{p}_{i} - j$ is the location of the path after first j steps. Let H _n,k be the span of all these paths, each treated as a basis state. Denote ${p}_{<i} = {p}_{1}\ldots {p}_{i-1}$ and $p_{\geq i} = p_i \ldots p_{n}$.

Fig. 9

The graph L _k−1.

To define Φ, it is enough to define its action on each E _i , done as follows, where ${\lambda }_{j} =\sin (\,j\pi /k)$.

$$\Phi ({E}_{i})\vert {p}_{<i}00{p}_{\geq i+2}\rangle = 0$$

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$$\Phi ({E}_{i})\vert {p}_{<i}01{p}_{\geq i+2}\rangle = {{{\lambda }_{{z}_{i}-1}} \over {{\lambda }_{{z}_{i}}}} \vert {p}_{<i}01{p}_{\geq i+2}\rangle + {{\sqrt{{\lambda }_{{z}_{i } +1 } {\lambda }_{{z}_{i} -1}}} \over {{\lambda }_{{z}_{i}}}} \vert {p}_{<i}10{p}_{\geq i+2}\rangle$$

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$$\Phi ({E}_{i})\vert {p}_{<i}10{p}_{\geq i+2}\rangle = {{{\lambda }_{{z}_{i}+1}} \over {{\lambda }_{{z}_{i}}}} \vert {p}_{<i}10{p}_{\geq i+2}\rangle + {{\sqrt{{\lambda }_{{z}_{i } +1 } {\lambda }_{{z}_{i } -1}}} \over {{\lambda }_{{z}_{i}}}} \vert {p}_{<i}01{p}_{\geq i+2}\rangle$$

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$$\Phi ({E}_{i})\vert {p}_{<i}11{p}_{\geq i+2}\rangle = 0$$

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Theorem 7 If $d = 2\cos (\pi /k)$, then Φ ∘ ρ _A is a unitary representation of B _n in r-dimensional vector space.

Now define the subspace H _n,k,l of H _n,k by ${H}_{n,k,l} = {\rm span}\{\vert p\rangle\!\!: {z}_{n}(p) = l\}$. Define Tr _n as in Eq. 35. We are ready to show the algorithm after some final comments: The Fact in Step 2 is covered by the standard Hadamard Test on both the real and the imaginary parts; see Aharonov et al. (2006b) for details. For each basis σ _i , it is not hard to check that Φ ∘ ρ _A (σ _i ) can be implemented in polynomial time on a quantum computer. The algorithm for approximating the Jones polynomial on B ^tr is as follows. The averaging over polynomial number of samples at the last step can be shown to give enough approximation by the standard Chernoff bound.

Algorithm 1 Approximate Jones trace closure
1. Repeat for j = 1 to t = poly(n, m, k):
a. Classically, pick a random path p ∈ P n,k with probability Pr(p) ∝ sin(πl ∕ k), where l is the index of the site at which p ends.
b. Use Hadamard Test to output a random variable x j with E[x j ] = Re〈p|Q(B)|p〉.
2. Use Hadamard Test to output a random variable y j with E[y j ] = Im〈p|Q(B)|p〉.
3. Let $r = {1 \over t} \sum\nolimits_{j}({x}_{j} + i{y}_{j})$. Output ${(-A)}^{3w({B}^{\rm tr}) }{d}^{n-1}r$.

The case of plat closure can be reduced to the trace closure case by the following observation. The plat closure of a braid B is isotopic to the trace closure of C, obtained by putting B on top of n/2 capcups, where a capcup is a cup on top of a cap. The algorithm is largely the same as the one for trace closure, except that one needs an observation that the state $\vert\alpha\rangle = \vert 1,0,1,0,\ldots,1,0\rangle$ can be used to connect the Jones polynomial of the plat closure and the trace function. To be more precise, one has

Fact 3 $\langle \alpha \vert \Phi \circ {\rho }_{A}\vert \alpha \rangle = {{N} \over {\sin (\pi /k){d}^{n/2}}} T{r}_{n}(\Phi \circ {\rho }_{A}(C))$.

By this, it is enough to estimate 〈α|Φ ∘ ρ _A |α〉, which can be done by Hadamard test again like in the trace closure case. The algorithm is as follows.

Algorithm 2  Approximate Jones plat closure
1. Repeat for j = 1 to t = poly(n, m, k):
a. Generate the state $\vert\alpha\rangle = \vert 1,0,1,0,\ldots,1,0\rangle$.
b. Use Hadamard Test to output a random variable x j with E[x j ] = Re〈α|Q(B)|α〉.
2. Use Hadamard Test to output a random variable y j with E[y j ] = Im〈α|Q(B)|α〉.
3. Let $r = {1 \over t} \sum\nolimits_{j}({x}_{j} + i{y}_{j})$. Output ${(-A)}^{3w({B}^{\rm tr}) }{d}^{3n/2-1}r\sin (\pi /k)/N$.

4.3 The PromiseBQP-Hardness of Approximating Jones Polynomials

A brief discussion of the idea in Aharonov and Arad (2006) of simulating a quantum circuit by an oracle U approximating the Jones polynomials will be presented here. First, based on a simple procedure similar to the one in Sect. 3.1.2, it is enough to approximate 〈0|U|0〉 for any polynomial size quantum circuit U. So we want to efficiently construct a braid B such that B has polynomially many crossings and 〈α|Φ ∘ ρ _A (B)|α〉 ≈ 〈0|U|0〉. (Here the ≈ sign means approximation to any desired accuracy.)

Note that in this approach, the working space has to be encoded by paths on graph L _k−1. Thus we need to encode the space for the original circuit U by the path space. A simple encoding uses the following four-step paths:

$$\vert 0\rangle \rightarrow \vert 1010\rangle,\quad\vert 1\rangle \rightarrow \vert 1100\rangle$$

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Suppose U is decomposed as U = U _m …U ₁, where each U _i is an elementary gate acting on at most two qubits. It can be assumed without loss of generality, that each U _i operates on adjacent qubits. Using the path encoding, U _i operates on eight qubits. Note that the path is not arbitrary in Q _{4n, k} since it always returns to the original point every four steps. Denote by S the subspace spanned by all these paths; then it is sufficient if one can efficiently find B _i ∈ B _4n such that Φ ∘ ρ _A (B _i ) ≈ U _i on the subspace S to the polynomially small accuracy. It turns out that it is doable, as the following density theorem shows for B ₈.

Theorem 8 Suppose $\,\tilde{\!U}$ is an encoded two-qubit quantum gate, then for any δ > 0 and k ≥ 11, one can find a braid B ∈ B ₈ with poly(k,1/δ) generators of B ₈ such that $\|(\Phi \circ {\rho }_{A}(B)\vert p\rangle - (\,\tilde{\!U})\vert p\rangle \| \leq \delta$ for all |p〉∈ H _n,k,1.

The proof of this theorem is the core technical part needed to show the hardness, but it is a bit far away from the PromiseBQP-completeness notion that is being discussed. The readers are referred to Aharonov and Arad (2006) for the details.

4.4 Additive Approximation of the Tutte Polynomials

The results for the Jones polynomials in Aharonov et al. (2006b) and Aharonov and Arad (2006) are generalized to the Tutte polynomials in Aharonov et al. (2007a). The multivariate Tutte polynomial is defined as follows.

Definition 18 (Tutte polynomial) Given an undirected graph G = (V, E) with a weight function on edges v = {v _e : e ∈ E}, the Tutte polynomial is

$${Z}_{G}(q,{\bf v}) = \sum\limits_{A\subseteq E}{q}^{k(A)} \prod\limits_{e\in A}{v}_{e}$$

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where k(A) is the number of connected components in the subgraph (V, A).

When all v _e 's are equal to the same v, then the polynomial becomes

$$Z_G(q,v)=\sum\limits_{A\subseteq E}q^{{k(A)}}v^{\vert A\vert},$$

which is essentially the same as the standard Tutte polynomial

$${T}_{G}(x,y) = \sum\limits_{A\subseteq E}{(x - 1)}^{k(A)-k(E)}{(y - 1)}^{\vert A\vert +k(A)-\vert V \vert }$$

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under the change of variables

$$x = 1 + q/v,\quad y = 1 + v$$

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$$q = (x - 1)(y - 1),\quad v = y - 1$$

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The main results in Aharonov et al. (2007a) are as follows. First, there is an efficient algorithm to additively approximate the multivariate Tutte polynomial of a given weighted planar graph. Second, there exists a range of complex weights and complex values of q such that the additive approximation of the multivariate Tutte polynomial at those points to within some scale is PromiseBQP-hard. The approximation windows in the above two results do not match that nicely in the non-unitary case, in which they modified the definition to get the PromiseBQP-complete problems.

The rest of this section will mainly illustrate some ideas of the algorithm. The algorithm actually takes an approach similar to that the Jones polynomial, with generalizations of various objects. First, we start at a planar graph rather than a braid. Like using closures of braids, we also change the planar graph to a knot-like object. Here the medial graph is used. For a planar graph G, the medial graph L _G is obtained in the following way. First encircle the facets of G with lines, and then cross the lines that surround each edge by putting a vertex in the middle of the edge. See Fig. 10 for an illustration. Note that the resulting graph L _G is 4-regular.

Fig. 10

Medial graph of a planar graph.

The regions of the medial graph can be {black,white}-colored so that no two adjacent regions have the same color. This coloring is unique up to an overall flip; let the outer region be fixed to be white.

For each crossing, there are two ways to break it, depending on whether one connects the two black or the two white regions. Let σ be an m-bit string indicating the crossing-breaking choices, where $m = \vert E\vert = \vert \sigma \vert$ is the number of the crossings. Denote by Black(σ) the set of edges, the crossings corresponding to which are broken by σ with two black regions connected. The Kauffman bracket of the medial graph L _G can now be defined by

$$\langle {L}_{G}\rangle = \sum\limits_{\sigma }{d}^{\vert \sigma \vert }\prod\limits_{e\in {\rm Black}(\sigma)}{u}_{e}$$

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where the new variables u _e and the old ones v _e are related by v _e = dv _e . The following fact connects the Kauffman bracket of L _G and the Tutte polynomial of G.

Fact 4 $\langle {L}_{G}\rangle (d,{\bf u}) = {d}^{-n}{Z}_{G}({d}^{2},d{\bf u})$.

Thus it is enough to calculate the Kauffman bracket of the medial graph. To this end, one needs a generalized Temperley–Lieb algebra GTL(d ), where one allows an arbitrary number of strands and allows cases with different numbers of upper and lower pegs in one diagram. A diagram does not “change” by adding some trivial strands; that is, those going from an upper peg directly to a lower peg without crossing any other strand. With these relaxations, the product can be defined similarly to the Jones polynomial case (by putting one on top of the other) in a consistent manner.

Using this bridge, generalized versions of the mapping Φ and the path model representation of GTL(d ) can be defined. But now the representative does not need to be unitary. Again the homomorphism property of representation reduces the task to approximating each basic tangle diagram.

There is one catch, however: The final scale of the approximation window depends on the decomposition of L _G into some simple structures called basis tangles. The optimal decomposition is not known to be easy to obtain. This is also a drawback of the algorithm: the approximation window contains a quantity which is complicated and not directly about the graph itself (but about the layout of it on the two-dimensional plane).

5 Concluding Remarks

5.1 Some Other PromiseBQP-Completeness Related Problems

In Wocjan and Zhang (2006), it is shown that the problem of local unitary phase sampling is also PromiseBQP-complete, as is a problem called local unitary average eigenvalue. Even earlier, Knill and Laflamme found that the quadratically signed weight enumerators problem is also PromiseBQP-complete (Knill and Laflamme 2001).

In the track of the Jones polynomial and Tutte polynomial, Wocjan and Yard gave new quantum algorithms for approximating HOMFLYPT two-variable polynomials of trace closures of braids (Wocjan and Yard 2008). They also gave algorithms for approximating the Jones polynomial of a general class of closures of braids.

There are some equivalent models of efficient quantum computation, such as adiabatic quantum computation (Aharonov et al. 2004) and one-way quantum computation (Browne and Briegel 2006). One can also view the canonical problems in those models as complete problems for the standard quantum Turing machine or quantum circuit model.

One can interpret the problem of SMPE in Sect. 3.2 as approximating the total weight of cycles of length t passing a vertex j on a weighted graph. The work (Janzing and Wocjan 2007) can actually show the PromiseBQP-completeness even for Hamiltonians with {1, −1, 0} entries. However, allowing the weight to be −1 makes weights on cycles cancel, giving the problem a quantum flavor. Childs recently studied the problem with the weight to be chosen only from {0, 1} (Childs 2009).

5.2 Future Directions

There are two issues in the state of the art of PromiseBQP-completeness. One is that all the known PromiseBQP-complete problems are not “natural” enough. In some sense, they are all describing the same class using different languages, though the difficulty of translation may be at different levels. One can say that all completeness results have this feature, but the key reason why NP-completeness is so important is that there are so many natural and seemingly unrelated combinatorial problems in theoretical computer science, discrete mathematics, and various other branches of mathematics and science. But not many natural PromiseBQP-complete problems are known so far.

Another direction is to try to use the PromiseBQP-complete problem to study the classes PromiseBQP and BQP. For example, one of the main open questions in quantum complexity theory is whether BQP is in PH, the polynomial hierarchy. The current best known upper bound of BQP is AWPP, a not-so-natural subclass of PP. (See the textbooks (Arora and Barak 2009; Goldreich 2008; and Papadimitriou 1994) and the “complexity zoo” (currently at http://qwiki.stanford.edu/wiki/Complexity_Zoo) for definitions of these complexity classes.) Could the known PromiseBQP-complete problems shed any light on the open problem?