# QIP 2012 Day 4

Possible location of QIP 2014??

### Aleksandrs BelovsSpan Programs for Functions with Constant-Sized 1-certificates, based on 1105.4024

Consider the problem of finding whether or not a triangle exists in a graph with n vertices. Naively using Grover’s algorithm requires time $n^{3/2}$. But a series of improvements have brought this down to $n^{10/7}$ (using Grover with better combinatorics) and $n^{13/10}$ (using element distinctness in a clever way). The current paper reduces this run time to $n^{35/27}$. That is, the exponent is improved from 1.3 to 1.296296296…
Before you roll your eyes and move to another tab, hear me out. The real improvement is in the technique, which is radically new, and has applications to many problems; not only triangle-finding.
The idea is to construct something called a learning graph. Each vertex is a subset of [n], and (directed) edges are weighted, and correspond to (sets of) queries. These are like randomized decision trees except they can have loops.
The complexity of the resulting algorithm (which we’ll describe below) is a little tricky to define. There is a “negative complexity” which is simply $sum_e w_e$. And a positive complexity which involves maximizing over all yes inputs $x$, and then minimizing over all flows of size 1 where the source is the empty set and the sinks are the 1-certificates for $x$.
Algorithms in this model correspond to flows of intensity one. The vertex corresponding to the empty set is the root, and 1-certificates (i.e. proofs that x is indeed a yes instance) are the sinks. The cost is $sum_e p_e^2/w_e$, where $p$ denotes the flow.
From a learning graph, there is a recipe to produce a span program, and in turn a quantum query algorithm, with the query complexity equal to the geometric mean of the negative and positive complexities of the learning graph.
As an example, consider the OR function. Here the learning graph is comprised entirely of a root and n leaves. Weight each edge by 1, so the negative complexity is n. If there are k solutions, then the resulting flows are 1/k on each edge, and so the positive complexity is 1/k. The algorithm’s complexity is $sqrt{n/k}$. Phew!
Less trivial is element distinctness:
Are there two indices $ineq j$ such that $x_i=x_j$?
Again the algorithm achieves the optimal complexity, which is $O(n^{2/3})$, matching Ambainis’s algorithm.
There are lots of new applications too, to improving the triangle runtime, but also getting a bunch of polynomial speedups for other problems that either are, or resemble, subgraph containment. For example, it improves some of the results from 1011.1443, and gets linear time algorithms to check whether a graph contains subgraphs from certain families, like lines, T-shaped things, and lines that go through stars.

### Francois le Gall, Improved output sensitive quantum algorithms for Boolean matrix multiplication

Boolean matrix multiplication means that the matrices are binary, + is replaced by OR, and * is replaced by AND. This arises in many cases, often related to graph theory, such as computing the transitive closure of a graph, triangle-detection, all-pairs shortest path, etc. Also it has application to recognizing context-free languages.
The naive algorithm is $O(n^3)$. Reducing to integer multiplication gives $O(n^{2.3ldots})$. Quantumly, we can use Grover to compute each entry, for a total runtime of $O(n^{2.5})$.
But suppose C has only one nonzero entry. Then we can do recursive Grover search and get time complexity $O(n^{1.5})$ [Buhramn Spalek ‘05].
If it has $l$ nonzero entries, we get $O(n^{1.5}) sqrt l$, neglecting polylogs.
The parameter l represents the sparsity of matrix “output sensitive algorithms”.
In this work, the runtime is improved to $O(n^{3/2} + nl^{3/4})$. There is also a more complicated result for query complexity. The techniques are adapted from papers of Williams & Williams, and Lingas. They also make use of a wide range of polynomial speedup tricks: Grover, approximate counting, triangle-detection, etc. The talk leaves the exciting impression that we not yet finished seeing improvements to these techniques.

### Richard Cleve Discrete simulations of continuous-time query algorithms that are efficient with respect to queries, gates and space, joint work with Dominic Berry and Sevag Gharibian.

After he was introduced, Richard sheepishly admitted, “The title isn’t really that efficient with its use of space.”
Normally the oracle for a quantum query algorithm defines the oracle query by flipping the phase of certain bits. In the continuous-time setting, we can get other phases. So, for example, a ”half-query” would give $Q |x rangle = i^{x_j} | x rangle$, since i is a square root of minus one. In this model, we only charge 1/2 of a query for this oracle call. More generally, we can have arbitrary fractional queries. Then we can take a limit and get to continuous time to find $expbigl(i(H_U + H_Q)tbigr)$ where $H_U$ is the driving (gate) Hamiltonian and $H_Q$ is a query Hamiltonian.
It is essential to be able to translate from the continuous-time model back to the discrete-time model for algorithmic applications. (cf. The work on NAND trees.) Is there a systematic way to do this translation? Yes, CGMSY-M 09 (improved by LMRSS 11) showed that the query complexity can be directly translated with the same number of queries. But these constructions are not gate efficient. Can we improve the gate complexity of the translation?
We have to be careful because the gate complexity will depend on the complexity of the driving Hamiltonian, so we have to carefully define a notion of approximately implementable Hamiltonian. If we have an epsilon-approximate implementation, and a driving Hamiltonian H, then the number of gates is at least $G ge log (|H|)/epsilon$, where the norm is the operator norm of the Hamiltonian (but modulo some details about the global phase.)
Main results: We can translate from the continuous-time setting to the discrete-time setting by using:

 $O(T log T / log log T)$ queries $O(T G log^2 TG)$ gates $O(log^2 TG)$ qubits of ancilla space

### Miklos Santha, Hidden Symmetry Subgroup Problems, joint work with Thomas Decker, Gábor Ivanyos and Pawel Wocjan

First let’s observe an interesting connection. In the dihedral group $D_{2p}$ for a prime p, the two-element subgroups generated by reflections coincide with the symmetries of the level sets of quadratic polynomials over $mathbb{F}_p$.
Can we more generally study subgroups which are hidden by symmetries? This generalizes and connects many problems in quantum algorithms, e.g. the hidden subgroup problem (HSP) over non-abelian groups, non-linear hidden structures and hidden polynomial problems among others, as the above example demonstrates.
Recall that the HSP is defined as follows. There is a function (for which you have an oracle) which is constant and distinct on the cosets of a subgroup of some group G. You want to obtain a generating set for the subgroup.
The hidden symmetry subgroup problem (HSSP): the input is a group G, a group action $circ$ which acts on a set M, a closed set of subgroups $mathcal{H}$, and there is an oracle function $f:M to S$, such that (we’re promised) for some subgroup $H in mathcal{H}$, $f(x) = f(y) leftrightarrow H circ x = H circ y$ for $x,y in M$. You want to output this (unique) H.

### Rahul Jain and Ashwin Nayak: A quantum information cost trade-off for the Augmented Index

Consider a twist (due to Yao ‘82) on the usual communication complexity problem. Alice has x, Bob has y, and they both want to compute f(x,y). But this time, they don’t want to leak (to the other party) more information than is already expressed by the answer, e.g. the millionaire problem.
Start with the usual example: equality testing. There is a O(log n) one-way protocol with 1/poly(n) error, and reveals only O(1) bits about one input.
Augmented index: Alice has $x_1, ldots, x_n$. Bob has $k, x_1,ldots, x_{k-1}, b$.
They want to know whether $b=x_k$.
(This has many connections to other works in theoretical computer science, not all of which are obviously communication-related. See paper for detail.)
Main result: For any quantum protocol which computes the augmented index with probability $1-epsilon$ on the uniform distribution, we must have either: 1) Alice reveals $Omega(n/t)$ bits of info about x, or 2) Bob reveals $Omega(1/t)$ about k, where t is the number of messages. This is true even when restricted to 0-inputs.
There is also an earlier classical result by the same authors: Alice reveals $Omega(n)$ or Bob reveals $Omega(1)$. If there are only two messages, then the lower bound for Bob is improved to $Omega(log n)$ (which is optimal), even when restricted to 0-inputs.
At this point, Ashwin, apparently reading the audience’s mind, puts up a slide that is not quite as pertinent as Josh Cadney’s opening question, but is still what we are all thinking:
“Why Augmented Index? Why privacy w.r.t. 0-inputs?”
Answer: Suppose we care about a streaming algorithm, which for quantum is relevant because we might only have O(1) qubits for the foreseeable future. There are exponential space improvements in this model, but let’s just say they don’t look like the sort of problems that Cisco was dying to implement on their routers. Augmented Index, by contrast, is much more natural. One semi-related version is Dyck(2), which is determining whether a sequence of (,),[,] is properly bracketed (this is a special case of recognizing context-free languages). Here we observe a big separation between one and two passes over the input (if the second pass goes in reverse order), and the lower bound on the (classical) one-pass complexity is based on (classical) lower bounds for the augmented index problem.
Having motivated the question, Ashwin is left with “-2 minutes” to finish his talk. Talk about a cliffhanger! Now he’s guaranteed to have the proof accepted to QIP 2013 (although he did give away the punchline by stating the theorem at the start of his talk). Moreover, the chair apparently has a sign for -2 minutes left, since he didn’t say anything, he just held up a sign. I guess staying on time is not a strong point in our community. 🙂
Ashwin is now ruining his cliffhanger, giving an intuitive and cute overview of the proof! He claims that this part of the talk is “in reverse.”

### Sevag GharibianHardness of approximation for quantum problems, joint work with Julia Kempe

Finding the ground state energy of a local Hamiltonian is QMA-complete, and this is harder than NP, because, um, there’s quantum in there too, and well, it seems harder? Sev wants to say something more concrete here. We’ve given up on solving local Hamiltonian exactly in general, because we’ve already given up on solving NP on a quantum computer (mostly). But surely there are good approximation algorithms for these things. Right? Well, today, we’ll talk about hardness of approximation, thus casting complexity-theoretic cold water on seemingly less ambitious goals.
All we know about approximating the local Hamiltonian problem:

• PTAS on planar graphs
• $1/d^{k-1}$-approximation algorithm for dense graphs with k-local Hamiltonians.

This talk takes a step away from QMA, and considers instead the complexity class cq-$Sigma_2$, which is the set of problems that can be expresses as “there exists x, such that for all $|psirangle$, a poly-time verifier V accepts $(x, |psirangle)$.
Their main result is to show that the following problem is complete for cq-$Sigma_2$, and the hardness part holds even if we demand only approximate solutions.
The quantum succinct set cover problem: Given a set of 5-local Hamiltonians acting on n qubits, and a parameter $alpha>0$, and the promise that
$sum_{iin S} H_i succeq alpha I$,
find the smallest subset $S'subseteq S$ such that
$sum_{iin S'} H_i succeq alpha I$.
The diagonal case was shown to be hard to approximate by Chris Umans in 1999.
For the technical details, there are lots of nice tools, and you’ll just have to look at the paper (when it comes out…).

### Jérémie Roland,Quantum rejection sampling, joint work with Maris Ozols and Martin Roetteler

Let’s first review classical rejection sampling, which is a tool that you can use to sample from arbitrary probability distributions. It has numerous applications: Metropolis sampling, Monte Carlo algorithms, optimization algorithms (e.g. simulated annealing) and many others. In the quantum case, we wish to replace probabilities by amplitudes.
Consider two probability distributions P and S. Suppose we can sample from P repeatedly, but we wish to sample from S. Define $Pr[{rm accept} k] = gamma s_k/p_k$ and continue to sample until you accept. Here we will choose the parameter $gamma$ in a way to try to minimize the expected amount of sampling from P that we need to do. The best way to do it is to choose $gamma = min_k p_k/s_k$. Then the expected number of samples from P to get a sample from S is $1/gamma$.
Now for the quantum case. We are given access to a black box which prepares a state $|pi^xirangle = sum_k pi_k |xi_krangle |krangle$ where the $pi_k$ are known amplitudes. We want to prepare the state $|sigma^xirangle = sum_k sigma_k |xi_krangle |krangle$ in terms of some new amplitudes. What is the randomized query complexity of this?
Instead of computing a function, which is the normal thing that we use oracles for, we wish to generate a quantum state. There is an oracle, which is a unitary that hides the label $xi$ in a non-explicit way. The action of the oracle is $O_xi |0rangle = sum_k pi_k |xi_krangle |krangle$.
Let’s define quantum state preparations:
We have a set of quantum states $Psi = {psi_xi}$
set of oracles $O = { O_xi }$
Goal: generate $psi_xi$ given black box access to $O_xi$
What’s the minimum number of calls to the oracle to get the correct state (up to a coherent error with amplitude $sqrt{epsilon}$)?
We introduce a coherent coin.
$O_xi |0rangle to sum_k pi_k |xi_krangle |krangle (sqrt{ pi - alpha})|0rangle + alpha|1rangle)$
This rotated ancilla is like a classical coin, but now we have a superposition. If we measure the ancilla then we accept if we get one.
Using amplitude amplification, we can get a speedup over the naive accept probability.
We can also optimize over the rotation angle to get the optimum. This can be done efficiently since it’s an SDP. They can prove that this is an optimal algorithm. [photo 70]
Now for the applications. Linear systems of equations: the HHL algorithm basically just does phase estimation followed by quantum rejection sampling. [photo 71] The quantum Metropolis algorithm is a way to prepare a Gibbs state of H at inverse temperature beta.
Direct mapping of Metropolis MRRTT algorithm OK for diagonal Hamiltonians, but if Hamiltonian is not diagonal we want to avoid the work of diagonalizing it.
One of the reasons that the Q. Metropolis algorithm was difficult was that the “reject” moves required that you go back to a state that you just measured. You can avoid having to deal with this difficulty if you use quantum rejection sampling. One can amplify the accepted moves and therefore avoid having to revert moves at all!
Boolean hidden shift. f(x+s) = f(x) find s in Z_2^n. If the function is a delta function, then you can basically only do Grover. You can do bent functions (those with flat Fourier spectrum) with only 1 query [Roetteler 10]. These are the extreme cases, but what happens when we interpolate between them? We can use quantum rejection sampling to analyze this problem.
Other applications: Amplify QMA witnesses [MW05, NWZ09], prepare ground states of PEPS parent Hamiltonians on a quantum computer [STV11]. [photo 76]

### Rolando Somma, Spectral Gap Amplification, joint work with Sergio Boixo

Preparation of eigenstates is a central problem and fast quantum methods for computing expectation values of observables leads to speedups for many problems. Let’s begin by considering adiabatic state transformations. The run time is governed by the spectral gap of the interpolating Hamiltonian. The generic cost C(T) of a quantum algorithm that prepares the eigenstate depends on the inverse power of the spectral gap.
given H with eigenstate psi and gap Delta, can we construct an H with same eigenstate psi but a larger gap? (without cheating by multiplying by some constant?) We want to consider that the Hamiltonian is a sum of terms which are, say, local. We have a black box which on input k, s evolves according to $e^{-i s H_k}$. How many times do we need to use this box? We require that the cost of evolving with H’ is the same.
They achieved a quadratic gap amplification for the following case: if H is frustration free, then the new gap is $Omega(sqrt{Delta/L})$ where L is the number of terms in the Hamiltonian. In particular, when the spectral gap is very small in terms of L this gives an improvement. It turns out this amplification is optimal.
Question: What prevents you from iterating amplification process?
Ans. The new Hamiltonian isn’t FF, so you can only do the amplification once.
S. Flammia: Optimality of results?
Ans. There may be better constructions with respect to the scaling in L.

### Rajat Mittal, Quantum query complexity for state conversion, joint work with Troy Lee, Ben Reichardt, Robert Spalek and Mario Szegedy

We live in a quantum world, so why are we always trying to solve classical tasks, like computing f(x), for x specified by an oracle? How about trying to prepare $|psi(x)rangle$ given x specified by an oracle? Or even more generally, converting $|psi_1(x)rangle$ into $|psi_2(x)rangle$? To make things simple, let’s focus on creating $|psi(x)rangle$, although the results apply to more general state conversion problems as well. The difficulty of doing this is defined entirely by the Gram matrix $G_{x,y} = langle psi(x)|psi(y)rangle$.
Previous work on adversary bounds gave a tight characterization of quantum query complexity for boolean functions. The present work extends the result to general functions, and indeed to state conversion.
$gamma_2(G-J|Delta)$
Ok, what is this thing $gamma_2(M|Delta)$?
Without the $Delta$, we define $gamma_2(M)$ to be the minimum over factorizations ${u_x}$, ${v_y}$ (meaning that $M_{x,y} = langle u_x, v_yrangle$ of $max |u_x|,|v_y|$
w.r.t. $Delta$; it’s similar, but with $M_{x,y} = langle u_x, v_yrangle N_{x,y}$.
e.g. $gamma_2(M|J) = gamma_2(M)$ and $gamma_2(M|M)=1$.
This gives some nice corollaries, e.g. that function composition does what you’d expect to query complexity.

### Fernando Brandao, Random quantum circuits are unitary polynomial designs, joint work with Aram Harrow and Michal Horodecki

Previously blogged here.
Questions:
Patrick Hayden: To fool a circuit takes more time than the circuit does. Can you introduce some kind of restriction to make it go in a more favorable direction?
Aram: We could try to fool quantum Turing machines, but I don’t think our techniques say much that’d be helpful there.
Renato: Could you replace a random circuit by a random Hamiltonian?
Ans. Interesting question. Hayden et al do something like that. More interesting is to…

The headline stat: 42 papers were accepted out of 198 submitted, meaning a 21% acceptance rate. (Louis also said the “real” rate was 29%.)
There were 485 reviews, of which 104 are from 72 external reviewers.
There were 198 = 144 (early) + 54 (late) poster submissions.
Of the 42 talks, there were 4 plenary, 9 long, 27 regular and 2 merged.
The budget breakdown is: rental+technical 55k, food+drinks 90k, personnel 30k, student support 10k for a of total 185k.
There were 259 participants of which 113 students.
Thus conference fees were 105k and there was 85k from sponsors. This leaves a surplus of 5k, presumably to cover damages during the rump session.
For the locals, the conference booklet explains how to get to the rump session. You take the metro, and as Louis said, “you will visit a lot of stations.” Note that this post will go online after the rump session.
Also, where is QIP 2013???
BEIJING! (specifically, Tsinghua University)
Local organizers are: Andy Yao (general chair), Amy Wang (organizing chair), Giulio Chiribella, Luming Duan, Kihwan Kim, Jian Li, Yaoyun Shi, Shengyu Zhang.
There is a new CQI at Tsinghua. You can guess what it stands for. Note that few permutations of those three letters remain.
Tentative dates are: Jan 21-25 or Jan 28-Feb 1, 2013.
A truly open question is: where will QIP 2014 be?
Two tentative proposals on the table are IBM Yorktown Heights and Barcelona.
But volunteers are still welcome! If you want the experience of hosting QIP, send an email to Louis Salvail soon.
Possible locations include a science museum, a university campus and… La Pedrera? Seriously? (See picture at the top of the post.)
Then there was general discussion about QIP.
Suggestions include

• not posting the long versions (at least without the authors confirming)
• Renaming QIP Workshop -> QIP Conference.
• Getting rid of the 3-page submissions? Here was the heated discussion. There always has to be a heated discussion about something.
• Dec vs Jan? The audience seemed to overwhelmingly prefer January (among those who cared). And yet here they are, all here in December!

## 2 Replies to “QIP 2012 Day 4”

1. Marcin Kotowski says:

“Ashwin is now ruining his cliffhanger, giving an intuitive and cute overview of the proof! He claims that this part of the talk is â€œin reverse.â€ ” – just to make the details right, Ashwin only said: “Perhaps now I should give my talk in reverse” (or sth equivalent), but did not claim that he actually did it while sketching the proof 😉

2. Juan Bermejo Vega says:

Thank you guys for the great work.
Just some silly typos in the rejection sampling post:
1) There is a latex line which is not correctly shown: $\latex Pr[accept k] = \gamma s_k/p_k$.
2) Also, there are references to pictures that seem not to be available anywhere ( [photo 70], [photo 71])… are they accesible? It would be great to see them.