I Introduction
The complexity of quantum computation is completely determined by the complexity of quantum circuit amplitudes. These amplitudes can encode the solution to computationally hard problems, such as Jones polynomials Aharonov et al. (2009), Tutte polynomials Aharonov et al. (2007), and matrix permanents Scheel (2004); Rudolph (2009)
. Unfortunately, quantum mechanics does not provide us with a method for directly measuring these amplitudes or their corresponding probabilities. We must instead infer approximations to them via repeated computations.
There is often a significant difference between the complexity of an exact evaluation of a function and an approximation to it. For example, in the case of the ferromagnetic Ising model, an exact evaluation of its partition function is #Phard. However, a relativeerror approximation can be achieved with a classical computer in polynomial time Jerrum and Sinclair (1993). Another interesting example is the Jones polynomial. Exactly computing the Jones polynomial is #Phard Jaeger et al. (1990). However, unlike the ferromagnetic Ising model, the Jones polynomial retains this complexity for relativeerror approximations Kuperberg (2009). It is known that, for the same class of Jones polynomials, computing additiveerror approximations is BQPhard Aharonov and Arad (2011). Therefore, it seems unlikely that quantum computers can produce relativeerror approximations of Jones polynomials in polynomial time.
We show that the complexity of evaluating relativeerror approximations of Jones polynomials can be used to bound the classical complexity of approximately simulating random quantum computations. Under the assumption that (1) the Polynomial Hierarchy (PH) does not collapse Papadimitriou (2003) and (2) the averagecase complexity of relativeerror approximations of the Jones polynomial matches the worstcase complexity over a constant fraction of random links (Conjecture V), we prove that random quantum computations cannot be classically simulated up to a constant total variation distance (Theorem V). This argument follows as a natural extension to those given for Instantaneous Quantum Polynomialtime (IQP) circuits Bremner et al. (2010, 2016) and for other classes of random quantum circuits Boixo et al. (2016), when combined with results on approximate designs Harrow and Low (2009); Brandão et al. (2016). Our results provide a straightforward relationship between the approximation of Jones polynomials and the complexity of random quantum computations.
Many quantum circuit classes can be associated with functions that are #Phard to evaluate up to a relative error. This feature has been used to construct arguments in favour of a separation between the power of classical and quantum computation (for a review on this topic see Ref. Lund et al. (2017) and Ref. Harrow and Montanaro (2017)
). While we do not believe that quantum computers can exactly evaluate such functions, they play a vital role in defining the complexity of sampling from the output probability distribution of quantum circuits. Terhal and DiVincenzo
Terhal and DiVincenzo (2004) first used this feature to bound the capability of classical computers to simulate constantdepth quantum computations. This was later extended to the problem of sampling from linear optical networks Aaronson and Arkhipov (2011) and IQP circuits Bremner et al. (2010).Aaronson and Arkhipov Aaronson and Arkhipov (2011) proved an important relationship between the complexity of approximate sampling and the averagecase complexity of relativeerror approximations to counting problems. They showed that the complexity of evaluating relativeerror approximations to matrix permanents can be used to bound the classical complexity of sampling from random linear optical networks up to a constant total variation distance — a notion of approximation that is realistic for quantum computation. They conjecture that (1) the averagecase complexity of the permanent of Gaussian matrices is #Phard and (2) the permanent of Gaussian matrices satisfies a certain anticoncentration bound. Assuming that these conjectures are true, they show that the existence of an efficient classical algorithm which can approximately sample from these networks would imply the collapse of the Polynomial Hierarchy Aaronson and Arkhipov (2011). A similar result was proven for IQP circuits Bremner et al. (2016) — extending this argument to the quantum circuit model under a different averagecase complexity conjecture, where the equivalent anticoncentration conjecture could be proven.
These sampling problems are not just a good candidate for proving a separation between classical and quantum computation, but also for providing experimental benchmarks Boixo et al. (2016); Harrow and Montanaro (2017). This has motivated the study of many other sampling problems. Each of these conjecture the equivalence of the averagecase and worstcase complexity of relativeerror approximations of a given function. These include: (1) the permanent of Gaussian matrices Aaronson and Arkhipov (2011), (2) the gap of degreethree polynomials over Bremner et al. (2016); Miller et al. (2017), (3) output probabilities of conjugated Clifford circuits Bouland et al. (2017), and (4) complextemperature Ising model partition functions over dense Bremner et al. (2016), sparse Bremner et al. (2017), and threedimensional models Boixo et al. (2016); Gao et al. (2017); Hangleiter et al. (2017).
These averagecase complexity conjectures are each associated with a class of quantum circuits. These quantum circuits are not thought to be universal for quantum computation, with the exception of the threedimensional Ising model case, but nonetheless become universal under postselection. Understanding the distinctions between these conjectures is essential for understanding the relationship between these classes of quantum circuits. However, resolving such conjectures would require nonrelativising techniques Aaronson and Chen (2016). We therefore expect this to be a hard open problem.
We consider the problem of sampling from random quantum computations that are distributed according to an approximate unitary design. We observe that these approximate unitary designs produce output probability distributions that satisfy an anticoncentration bound. This bound is used to prove that if there exists an efficient classical algorithm which can sample from these distributions up to a constant total variation distance, then Stockmeyer’s Counting Theorem (Theorem A) can be used to produce relativeerror approximations to a constant fraction of their output probabilities (Theorem II). This same observation has been used to establish arguments for the complexity of random quantum circuits Boixo et al. (2016); Hangleiter et al. (2017) and conjugated Clifford circuits Bouland et al. (2017).
We define a natural model of random links via the braid group. A random braid is generated by applying generators of the braid group uniformly at random. A random link is then the plat closure of a random braid. We show that the output probability amplitudes of random quantum computations are proportional to the Jones polynomial of a random link. Furthermore, we show that in the path model representation with or , random braids on strands of length form an approximate unitary design (Corollary V). This leads us to conjecture that it is #Phard to approximate the Jones polynomial, up to a relative error, on at least a constant fraction of random links (Conjecture V). This provides a natural conjecture for bounding the classical complexity of simulating random quantum computations.
This paper is structured as follows. In Section II, we provide an introduction to random quantum computations and approximate unitary designs. We then state our result on the classical simulation of random quantum computations. In Section III, we briefly introduce the theory of knots, braids, and the Jones polynomial. We review the relationship between Jones polynomials and quantum computing in Section IV. In Section V, we relate the complexity of random quantum computations to the complexity of approximating the Jones polynomial of random links. Finally, we conclude in Section VI with some remarks and open problems.
Ii Random Quantum Computations
A random quantum computation
is the action of (1) preparing an initial state, (2) applying a randomly chosen unitary matrix, and (3) measuring in the computational basis. This is equivalent to sampling from a probability distribution
, where is a randomly chosen unitary matrix. [] For a unitary matrix , we define to be the probability distribution over integers , given byIt is natural to consider unitary matrices drawn from the uniform distribution. The uniform distribution over the unitary group
is defined by the Haar measure, which is the unique translationinvariant measure on the group. Unfortunately, random unitary matrices drawn from the Haar measure cannot be implemented efficiently by a quantum computer as they typically require an exponential number of gates Knill (1995).For our purposes, it is important that the random quantum computations can be implemented efficiently. We achieve this by weakening the requirement that the unitary matrices are drawn from the Haar measure. Instead, we require only that the unitary matrices are drawn from a distribution that is close to the Haar measure.
A unitary design is a distribution over a finite set of unitary matrices which imitates the properties of the Haar measure up to the moment. For convenience, let be the set of polynomials homogeneous of degree in the matrix elements of and homogeneous of degree in the matrix elements of . [Unitary design Roy and Scott (2009)] A distribution over unitary matrices in dimension is a unitary design if, for any polynomial ,
[approximate unitary design] A distribution over unitary matrices in dimension is an approximate unitary design if, for any polynomial ,
Brandao, Harrow, and Horodecki Brandão et al. (2016) showed that local random quantum circuits acting on qudits composed of polynomially many gates form an approximate unitary design. Here, is a universal set of gates containing inverses with each composed of algebraic entries. [local random quantum circuit] At each time step, two indices, and , are chosen uniformly at random from and , respectively. The gate is then applied to the two neighbouring qudits and . [Brandao, Harrow, and Horodecki Brandão et al. (2016)] Fix . Let be a universal set gates containing inverses with each composed of algebraic entries. There exists a constant such that local random quantum circuits of length
form an approximate unitary design. We shall, therefore, restrict our attention to random quantum computations where the unitary matrices are drawn from an approximate unitary design. We are interested in a classical simulation of random quantum computations, for which we have the following result: [] Let be a unitary matrix distributed according to an approximate unitary design and let be its corresponding probability distribution. Suppose that there is a classical polynomialtime algorithm , which, for any , samples from a probability distribution , such that . Then, for any such that , there is an algorithm which approximates up to a relative error on at least a fraction of matrices.
We prove Theorem II and several supporting lemmas in Appendix A. Theorem II tells us that, if there exists an efficient classical algorithm which can approximately sample from any random quantum computation, then, there is an algorithm which can approximate up to a relative error for a fraction of matrices . Suppose that this algorithm solves a #Phard problem, then, by Toda’s Theorem Toda (1991), the Polynomial Hierarchy collapses to its third level.
[Toda Toda (1991)]
Iii Knots, Braids, and the Jones Polynomial
We now briefly introduce the theory of knots, braids, and the Jones polynomial.
[Knot] A knot is subset of points in that is homeomorphic to a circle.
Informally, a knot is a tangled strand of string with the open ends closed to form a loop. Much like the everyday knots that we use when we tie our shoelaces, ties, and so on — mathematical knots are exactly that, except that the open ends are fused together.
The most simple knot you can think of is the unknot, also called the trivial knot, which is a closed loop without a knot (Fig. 0(a)). Other examples of knots include the trefoil knot (Fig. 0(b)), and the figure eight knot (Fig. 0(c)).
We have seen how a knot is an embedding of a circle in . We can now generalise this idea by considering an embedding of multiple circles in .
[Link] A link is a finite disjoint union of knots . Each knot in the union is called a component of the link.
[Oriented link] An oriented link is a link in which each component is assigned an orientation.
We can now see that a knot is a link of only one component. The generalisation of the unknot to a link on components is called the unlink, which is a collection of unknots that are not interlinked. An example of a slightly more interesting link is the Borromean rings link (Fig. 2), which has the property that removing any single component of the link gives the two component unlink.
A important problem in knot theory is the link recognition problem — given two links are they the same? To answer this, we must first ask, what does it mean for two links to be the same?
[Link equivalence] Two links and are said to be equivalent if there exists a orientationpreserving homeomorphism so that .
Essentially, two links are equivalent if they can be deformed into one another. We can prove that two links are equivalent by producing a set of instructions that will deform one link into the other. However, proving that two links are not equivalent is much more difficult, as we would need to prove that no set of instructions exist.
Link invariants are an important concept in knot theory as they allow us to study the link recognition problem.
[Link invariant] A link invariant is a function from the set of links to some other set, such that the output of the function depends only on the equivalence class of the link.
[Jones polynomial Jones (1985)] The Jones polynomial is a link invariant, which assigns to each oriented link a Laurent polynomial in the variable .
The Jones polynomial is characterised by the skein relation and the normalisation that the Jones polynomial of the unknot .
[Skein relation] Given three links , , and that are identical, except for a local region where they differ according to Fig. 3, then the following skein relation holds
The skein relation is sufficient for a recursive computation of the Jones polynomial of a link. It follows that the Jones polynomial of a link can be computed in time exponential in the number of crossings. A classic result of Jaeger, Vertigan, and Welsh Jaeger et al. (1990) states that exactly computing the Jones polynomial of a link is #Phard except when is one of a few special points. Bordewich et al. Bordewich et al. (2005) showed that it is BQPhard to approximate the Jones polynomial up to an additive error. Kuperberg Kuperberg (2009) proved that it remains #Phard to approximate the Jones polynomial up to a relative error.
[Jaeger, Vertigan, and Welsh Jaeger et al. (1990)] Evaluating the Jones polynomial of a link is #Phard except when with when it can be evaluated in polynomial time.
We now introduce the theory of braids, which provides us with a convenient way to represent any link. [Braid] Let
Then, an strand braid is a collection of nonintersecting smooth paths in connecting the points in to the points in . Informally, a braid is a collection of strands of string that may cross over and under each other, and must always move from left to right. An example of a braid is given in Fig. 4.
The set of all braids on strands form an infinite group , generated by the generators and their inverses . The generator crosses the strand over the strand and its inverse crosses the strand under the strand. [Braid group] The braid group on strands is the group given by the Artin presentation
Each braid can be described by a braid word. [Braid word] A braid word is word on the set of generators and their inverses . The length of a braid word is the number of characters in the word.
We can connect the endpoints of any braid in a number of ways to form a link. For a braid with an even number of strands a natural way to do this is by the plat closure. [Plat closure] The plat closure of a strand braid is the link formed by connecting pairs of adjacent strands on the left and the right of the braid. The link that is formed by the plat closure of the braid is often denoted . Alexander Alexander (1923) showed that we can generate all possible links this way. We can, therefore, describe any link as the closure of a braid given by its braid word. [Alexander Alexander (1923)] Every link can be represented by the closure of some braid.
Iv The Jones Polynomial and Quantum Computing
Freedman, Kitaev, and Wang Freedman et al. (2002a) established a quantum algorithm for additively approximating the Jones polynomial at any principle root of unity in polynomial time. This algorithm was later formalised by Aharonov, Jones, and Landau Aharonov et al. (2009). Freedman, Larsen, and Wang Freedman et al. (2002b) proved that when is a principle nonlattice root of unity, i.e. or , the problem of additively approximating the Jones polynomial is universal for quantum computation. Aharonov and Arad Aharonov and Arad (2011) extended this result to values of that grow polynomially with the number of strands and crossings. [Aharonov and Arad Aharonov and Arad (2011)] Let be a principle nonlattice root of unity, and let be a braid. Then, the problem of additively approximating the Jones polynomial to within the same accuracy as the AharonovJonesLandau algorithm Aharonov et al. (2009) is BQPhard.
The AharonovJonesLandau algorithm is based on the path model representation of the braid group Jones (1983, 1985), which is unitary when is a principle root of unity. For an integer , the path model representation of the braid group
is defined on the vector space spanned by walks of length
, on a vertex path graph , which start and finish on the first vertex.To calculate the dimension of this vector space it is sufficient to count the number of walks of length on the graph . From a combinatorial perspective, the walks on the graph can be seen as Dyck paths of length , which never go above a height . It is well known that the number of Dyck paths of length is the Catalan number, which provides an upperbound for the dimension of the vector space. [Catalan number] The Catalan number is defined by
For ,
Proof.
The claim follows directly from Stirling’s approximation for factorials. ∎
In this representation, each braid is mapped to a unitary matrix composed of algebraic entries. These unitary matrices have the property that the expectation value is proportional, up to an efficiently computable factor, to the Jones polynomial of the plat closure of . Aharonov, Jones, and Landau Aharonov et al. (2009) showed that such representations can be implemented efficiently on a quantum computer.
In their construction, the unitary representation of each generator of the braid group acts on a subspace of the Hilbert space of qudits. The SolovayKitaev theorem Kitaev et al. (2002) guarantees that these unitary matrices can be implemented efficiently. An entire braid is implemented efficiently by applying the corresponding unitary matrix of each generator in the order of the braid word of .
V Random Quantum Computations and Random Links
We now relate random quantum computations and the Jones polynomial of random links. We define a random link to be the plat closure of a random braid. [Random braid] A random braid on strands is generated by uniformly at random choosing generators from the set . [Random link] A random link is generated by the plat closure of a random braid.
In the path model representation the generators of the braid group are mapped to unitary matrices . In this representation, a random braid is equivalent to a product of random matrices chosen uniformly at random from the set . Since each acts on a subspace of the Hilbert space of qudits, a random braid is equivalent to a local random quantum circuit, with the number of strands proportional to the number of qudits. When or these gates are universal for quantum computation.
In the path model representation with or , there exists a constant , such that random braids on strands of length
form an approximate unitary design.
Proof.
The proof follows from combining Theorem II with the fact that the dimension of the vector space in the path model representation is bounded from above by the Catalan number and that the local dimension is bounded from below by . ∎
In the path model representation with or , there exists a constant , such that random braids on strands of length
form an approximate unitary design.
Proof.
We now relate the classical simulation of random quantum computations and the complexity of approximating the Jones polynomial of random links. Fix . Let or be an integer, and its corresponding root of unity. Let be a random braid on strands of length . Let be the path model representation of , and let be its corresponding probability distribution. Suppose that there is a classical polynomialtime algorithm , which, for any , samples from a probability distribution , such that and assume that Conjecture V holds. Then, there is a algorithm for solving any problem in and by Toda’s Theorem the Polynomial Hierarchy collapses to its third level.
In the notation of Theorem V. For some , it is #Phard to approximate the Jones polynomial up to a relative error on at least a fraction of random braids.
Conjecture V is based on the averagecase complexity of relativeerror approximations of Jones polynomials. It is known that it is #Phard to approximate the Jones polynomial up to a relative error in the worst case Kuperberg (2009). Therefore, Conjecture V states that this worstcase hardness result can be extended to an averagecase hardness result.
Assuming that Conjecture V holds and the Polynomial Hierarchy does not collapse, Theorem V tells us that there is no efficient classical algorithm which can sample from any random quantum computation. This implies that random quantum computations can not be efficiently simulated by a classical computer.
It is worth noting that the path model representation is equivalent to the Fibonacci representation of the braid group Shor and Jordan (2008). Therefore, our results extend to the random braiding of Fibonacci anyons.
Vi Conclusion & Outlook
We have provided strong evidence that simulating random quantum computations is intractable for classical computers. Specifically, we have shown that if Conjecture V holds and the Polynomial Hierarchy does not collapse, then there is no efficient classical algorithm which can approximately sample from the output probability distribution of random quantum computations.
There are a number of natural problems that remain to be solved. The most obvious of which is to resolve Conjecture V. Unfortunately, we are unaware of any proof techniques which are capable of extending the worstcase hardness result to an averagecase hardness result. Moreover, the results of Aaronson and Chen Aaronson and Chen (2016) imply that any proof of this conjecture would require nonrelativising techniques.
Another natural problem is whether Corollary V can be strengthened to random braids of a shorter length. In Theorem V, the length of a random braid is determined by the requirement that in the path model representation it is distributed according to an approximate unitary design. Therefore, any improvement to this bound yields a stronger version of Theorem V. It is an open problem whether this bound can be improved.
It would also be interesting to adapt our results to other functions, such as Tutte polynomials Aharonov et al. (2007), TuraevViro invariants Alagic et al. (2010), and matrix permanents Rudolph (2009). These functions are all known to be #Phard to compute in the worst case and BQPhard to approximate up to an additive error.
Acknowledgements
We thank Scott Aaronson, Sergio Boixo, Adam Bouland, Gavin Brennen, Aram Harrow, Saeed Mehraban, Ashley Montanaro, Peter Rohde, and Marco Tomamichel for helpful discussions. MJB acknowledges support from the Australian Research Council via the Future Fellowship scheme (grant FT110101044) and as a member of the ARC Centre of Excellence for Quantum Computation and Communication Technology (CQC2T), project number CE170100012.
Appendix A Proof of Theorem Ii
We now prove Theorem II, which is restated below for convenience. Our proof requires several lemmas which we prove in the remainder of the section.
See II
Proof.
Lemma A tells us that, for any , there is an algorithm, which approximates , up to an additive error
with probability at least over the choice of . Combining this with Lemma A and setting , it follows that there is an algorithm, which approximates up to a relative error on at least a fraction of matrices . ∎
We now prove Lemma A, which relates the simulation of random quantum computations to approximating individual output probabilities. Our proof closely follows that of Lemma 4 from Ref. Bremner et al. (2016). Let be a unitary matrix distributed according to an approximate unitary design and let be its corresponding probability distribution. Suppose that there is a classical polynomialtime algorithm , which, for any , samples from a probability distribution , such that . Then, for any such that , there is an algorithm, which approximates , up to an additive error
with probability at least over the choice of .
Proof.
Define
For any , we can use Stockmeyer’s Counting Theorem (Theorem A) to obtain a relativeerror approximation to in ,
Then,
As approximates up to an error , it follows from Markov’s inequality and the approximate design condition (Lemma A) that, for any ,
Therefore,
with probability at least over the choice of . ∎
The proof of Lemma A requires a classic result of Stockmeyer Stockmeyer (1985), which allows us to approximately count in the Polynomial Hierarchy. [Stockmeyer’s Counting Theorem Stockmeyer (1985)] Let be a function, and let . Then there exists an algorithm, which outputs a value , such that
We now prove that unitary matrices distributed according to an approximate unitary design satisfy the anticoncentration bounds claimed in Theorem II. This was proven independently by Hangleiter et al. Hangleiter et al. (2017). Let be a unitary matrix distributed according to an approximate unitary design, then, for any unit vectors , and a constant , the following holds
Proof.
The proof of Lemma A
combines the PaleyZygmund inequality and the approximate design condition. The PaleyZygmund inequality bounds the probability that a nonnegative random variable is small in terms of its first and second moment. [PaleyZygmund inequality] If
is a random variable with finite variance, and if
, thenWe are interested in bounding the probability that the random variable is small. In the case of an exact unitary design the first and second moments of match those of the Haar measure. For an approximate design the approximate design condition bounds the distance of the first and second moments of from those of the Haar measure. [Approximate design condition Brandão and Horodecki (2013)] If is a unitary matrix distributed according to an approximate unitary design, then, for any unit vectors , and an integer ,
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