*(A guest post by Ashley Montanaro on quantum property testing, a blogpost on his survey. Thanks Ashley!) *

**Quantum property testing**

As readers of this blog will be aware, the field of property testing is a vibrant and active area of research, with hundreds of papers having been written covering a vast range of topics. One direction which has perhaps been underexplored in recent years is the extent to which quantum computers could be useful in property testing. In this blog post, I will briefly discuss some examples of this kind. This is all based on a recent survey of quantum property testing, which is joint work with Ronald de Wolf. For technical details about everything I discuss here, and for references, please see this paper.

There are several ways in which one could generalise the idea of property testing to the quantum world. Perhaps the most obvious way, and the one which I will discuss in this post, is to consider quantum algorithms for testing properties of classical objects, with the goal being to find algorithms which are significantly faster than any possible classical property tester. We use the standard framework of property testing; the only difference is that our tester is allowed to be quantum.

A generalisation in a different direction which I will not discuss here is to testing whether *quantum* objects of some sort (such as quantum states or operations) have some property. This idea can be split into two cases, the first of which can be thought of as *classical* testers for quantum properties. While at first sight this idea might not seem to make much sense, in fact it is quite natural. For example, imagine we are given access to a quantum system which is claimed to be in a particular state, and we would like to verify that this is indeed the case. Further imagine that the only access to the system we are allowed is to perform measurements of some form, receiving classical measurement data. The problem of performing this kind of verification has been of both theoretical and practical interest to both physicists and computer scientists. Another, and very general, case we could consider is finding *quantum* testers for quantum properties. Here ideas and intuition from classical property testing can help develop algorithms for purely quantum tasks. See the survey and references therein for much more about the testing of quantum properties in each of these cases.

**Quantum testers for classical properties **

Quantum computers offer the prospect of dramatically faster algorithms for certain problems than are possible for classical computers. Property testing is a natural target for such algorithms, for at least a couple of reasons.

First, many of the most famous quantum algorithms known (such as Shor’s algorithm and Grover’s algorithm) operate in the query complexity model. Here the measure of complexity which we try to minimise is the number of queries to bits of the input used to compute some function of the input. Precisely the same model is usually used in property testing.

Second, it is known that quantum computers can only achieve super-polynomial speedups in the query complexity model for computing *partial* functions; that is, solving problems where we are promised that the algorithm will never receive certain inputs. In a property-testing problem we have such a promise: namely, the input either has some property, or is “far” from having that property.

This gives us hope that quantum algorithms could provide super-polynomial speedups for property-testing problems. However, it is not obvious that the (rather specific) nature of the promise in property testing lends itself to quantum speedup; indeed, it was only in 2002 that the first separation between quantum and classical property testers was demonstrated by Buhrman et al. In fact, this work gave two separations: \(O(1)\) vs. \(\Omega(\log n)\) for testing subsets of Hadamard codewords; and \(O(\log n \log \log n)\) vs. \(\Omega(n)\) for a property-testing variant of Simon’s problem.

Following these initial results, there has been a steady stream of other efficient quantum testers for classical properties. These are often based on applying in the property-testing context a quantum algorithm or primitive which has been useful elsewhere. One simple, and yet sometimes overlooked, way to obtain a more efficient quantum property tester from a classical property tester is via the important primitive of amplitude amplification, which is a more efficient quantum variant of classical probability amplification. Imagine we have a classical property tester for some property \(P\) such that the tester has perfect completeness (i.e. accepts objects with property \(P\) with certainty), uses \(q\) queries, and rejects objects \(\epsilon\)-far from \(P\) with probability \(p\). Then, using amplitude amplification, we obtain a quantum tester which rejects objects \(\epsilon\)-far from \(P\) with *constant* probability using \(O(q/\sqrt{p})\) queries. (Classical probability amplification would give \(O(q/p)\) queries.) For example, this can be applied to the standard classical tester for linearity of boolean functions to obtain a quantum property tester with constant success probability using \(O(1/\sqrt{\epsilon})\) queries, a quadratic improvement.

Some other examples where quantum algorithms which originated elsewhere have been used as part of efficient quantum property testers include:

- An algorithm of Chakraborty et al. for a variant of testing periodicity of a sequence, where the quantum algorithm uses \(O(1)\) queries but any classical algorithm must make \(\Omega(n^{1/4})\) queries. The algorithm is based on Shor’s algorithm and the classical lower bound is based on generalising prior work of Lachish and Newman. Quantum testers for periodic functions on groups, and some other group-theoretic properties, had previously been given by Friedl et al..
- A quantum algorithm of Atici and Servedio for testing \(k\)-juntas (boolean functions depending on at most \(k\) variables) using \(O(k/\sqrt{\epsilon})\) queries, which is a somewhat better complexity than the (later) best known classical bound of \(O(k \log k + k/\epsilon)\). The algorithm is based on the primitive of sampling from the Fourier spectrum of a boolean function, one of the earliest algorithmic ideas in quantum computing. The original algorithm of Atici and Servedio had a linear dependence on \(1/\epsilon\), but this can be improved to a square root using amplitude amplification.
- The use of quantum counting by Bravyi et al. and Chakraborty et al. to give polynomial speedups for testing a number of properties of probability distributions: whether two distributions are equal, whether a distribution is equal to a fixed distribution, whether two distributions have disjoint support, etc.
- The use of Ambainis’s efficient quantum algorithm for element distinctness to give polynomial speedups for testing bipartiteness and expansion of bounded-degree graphs. The basic idea here is to use the algorithm to efficiently find collisions in short random walks on the graph.

The last of these highlights an important gap in our knowledge: could there exist an exponential quantum speedup for testing a property of graphs? It is known that for problems with “too much” symmetry, quantum algorithms can achieve only a polynomial speedup over classical algorithms in the query complexity model. Graph properties (being invariant under permutations of the vertices) have an intermediate level of symmetry, and it is not clear whether a super-polynomial speedup is achievable in this setting.

**Lower bounds**

As well as the development of new quantum algorithms, an integral part of many of the above works demonstrating quantum-classical separations is the proof of new lower bounds on the number of classical queries required for property testing. It is also of interest to prove lower bounds on the *quantum* query complexity of property testing. This can be a challenging task. Indeed, one of the most successful techniques for proving quantum lower bounds, the adversary method originally introduced by Ambainis, suffers from a “property testing barrier” which prevents its application to property testing problems. This technique has subsequently been generalised to the negative-weight adversary method, which overcomes this barrier and indeed completely characterises quantum query complexity. However, this bound can be difficult to apply in practice and I am not aware of any examples of its use in property testing.

A different technique which has been successfully applied to lower-bound the quantum query complexity of various property testing problems is the method of polynomials. Here the idea is to exploit the fact that the acceptance probability of any quantum algorithm making a small number of queries is a low-degree multivariate polynomial. By proving a lower bound on the degree of any bounded polynomial which takes a large value on inputs with property \(P\), and a small value on inputs far from having property \(P\), we get a lower bound on the number of queries required by any quantum algorithm which tests \(P\).

One example where this idea can be applied is the testing of properties which correspond to linear codes. It turns out that for any code whose dual code has minimum distance \(d\), we get an \(\Omega(d)\) quantum lower bound on the number of queries required to test membership in the code. One such case is the Reed-Muller code of order \(d\), or equivalently the set of degree-\(d\) polynomials over \(F_2\); this argument can be used to show that any quantum algorithm that tests the property of being a degree-\(d\) polynomial must make \(\Omega(2^d)\) queries. Another, and much more complicated, example of the use of this method is a lower bound by Ambainis et al. on the quantum query complexity of testing expansion in graphs, which rules out an exponential quantum speedup for testing this property.

**Open questions**

The field of quantum property testing is just getting started and there are a number of interesting open questions. In particular, the quantum complexity of testing many properties which have been studied classically is wide open. Some examples include monotonicity, decision tree complexity, and almost all properties of graphs. In some cases, we might hope to find significant (e.g. exponential) speedups over any possible classical tester. The quantum property testers which have been found so far are largely based on applying quantum algorithms which originated elsewhere; but we could also hope to develop entirely new algorithmic techniques to attack these problems.

As well as considering specific problems, there are more wide-ranging questions which could be addressed. Can we achieve significant quantum speedups for testing properties with a high degree of symmetry – and more generally, what can we say about the structure of properties which admit efficient quantum testers? Can we find better (or simpler) lower-bound techniques for quantum property testers? Blais et al. have proven classical lower bounds based on an elegant technique using communication complexity – can their ideas be extended to prove quantum lower bounds based on quantum communication complexity?

Thinking more widely still, the question of just what it is about the structure of certain problems which makes them amenable to quantum speedup is of great interest and far from being resolved. Property testing provides a lens with which to study this question, whether by finding new examples of quantum speedup (or the lack thereof), or by uncovering deeper structural features of properties which determine whether they have efficient quantum testers. In both cases, there is the potential for new developments in quantum property testing to significantly influence quantum algorithmics more generally.