Florian Speelman

Computing with a full memory: catalytic space

We define the notion of a catalytic-space computation. This is a computation that has a small amount of clean space available and is equipped with additional auxiliary space, with the caveat that the additional space is initially in an arbitrary, possibly incompressible, state and must be returned to this state when the computation is finished. We show that the extra space can be used in a nontrivial way, to compute uniform TC1-circuits with just a logarithmic amount of clean space. The extra space thus works analogously to a catalyst in a chemical reaction. TC1-circuits can compute for example the determinant of a matrix, which is not known to be computable in logspace. In order to obtain our results we study an algebraic model of computation, a variant of straight-line programs. We employ register machines with input registers x1,..., xn and work registers r1,..., rm. The instructions available are of the form ri ← ri±u×v, with u, v registers (distinct from ri) or constants. We wish to compute a function f(x1,..., xn) through a sequence of such instructions. The working registers have some arbitrary initial value ri = τi, and they may be altered throughout the computation, but by the end all registers must be returned to their initial value τi, except for, say, r1 which must hold τ1 + f(x1,..., xn). We show that all of Valiants class VP, and more, can be computed in this model. This significantly extends the framework and techniques of Ben-Or and Cleve [6]. Upper bounding the power of catalytic computation we show that catalytic logspace is contained in ZPP. We further construct an oracle world where catalytic logpace is equal to PSPACE, and show that under the exponential time hypothesis (ETH), SAT can not be computed in catalytic sub-linear space.

The garden-hose model

We define a new model of communication complexity, called the garden-hose model. Informally, the garden-hose complexity of a function f:{0,1}n x {0,1}n -> {0,1} is given by the minimal number of water pipes that need to be shared between two parties, Alice and Bob, in order for them to compute the function f as follows: Alice connects her ends of the pipes in a way that is determined solely by her input x ∈ {0,1}n and, similarly, Bob connects his ends of the pipes in a way that is determined solely by his input y ∈ {0,1}n. Alice turns on the water tap that she also connected to one of the pipes. Then, the water comes out on Alices or Bobs side depending on the function value f(x,y). We prove almost-linear lower bounds on the garden-hose complexity for concrete functions like inner product, majority, and equality, and we show the existence of functions with exponential garden-hose complexity. Furthermore, we show a connection to classical complexity theory by proving that all functions computable in log-space have polynomial garden-hose complexity. We consider a randomized variant of the garden-hose complexity, where Alice and Bob hold pre-shared randomness, and a quantum variant, where Alice and Bob hold pre-shared quantum entanglement, and we show that the randomized garden-hose complexity is within a polynomial factor of the deterministic garden-hose complexity. Examples of (partial) functions are given where the quantum garden-hose complexity is logarithmic in n while the classical garden-hose complexity can be lower bounded by nc for constant c>0. Finally, we show an interesting connection between the garden-hose model and the (in)security of a certain class of quantum position-verification schemes.

Quantum communication complexity advantage implies violation of a Bell inequality

Significance For many communication complexity problems the quantum strategies, distinguished by using Bell nonlocal correlations, provide exponential advantage over the best possible classical strategies. Conversely, for any Bell nonlocal correlations there exists a communication complexity problem that is solved more efficiently using the former. Despite many efforts, there were only two problems for which one could certify that any strategy that outperforms the classical one must harbor Bell nonlocal correlations. We prove that any large advantage over the best known classical strategy makes use of Bell nonlocal correlations. Thus, we provide the missing link to the fundamental equivalence between Bell nonlocality and quantum advantage. We obtain a general connection between a large quantum advantage in communication complexity and Bell nonlocality. We show that given any protocol offering a sufficiently large quantum advantage in communication complexity, there exists a way of obtaining measurement statistics that violate some Bell inequality. Our main tool is port-based teleportation. If the gap between quantum and classical communication complexity can grow arbitrarily large, the ratio of the quantum value to the classical value of the Bell quantity becomes unbounded with the increase in the number of inputs and outputs.

Instantaneous non-local computation of low T-depth quantum circuits

Instantaneous non-local quantum computation requires multiple parties to jointly perform a quantum operation, using pre-shared entanglement and a single round of simultaneous communication. We study this task for its close connection to position-based quantum cryptography, but it also has natural applications in the context of foundations of quantum physics and in distributed computing. The best known general construction for instantaneous non-local quantum computation requires a pre-shared state which is exponentially large in the number of qubits involved in the operation, while efficient constructions are known for very specific cases only. We partially close this gap by presenting new schemes for efficient instantaneous non-local computation of several classes of quantum circuits, using the Clifford+T gate set. Our main result is a protocol which uses entanglement exponential in the T-depth of a quantum circuit, able to perform non-local computation of quantum circuits with a (poly-)logarithmic number of layers of T gates with quasi-polynomial entanglement. Our proofs combine ideas from blind and delegated quantum computation with the garden-hose model, a combinatorial model of communication complexity which was recently introduced as a tool for studying certain schemes for quantum position verification. As an application of our results, we also present an efficient attack on a recently-proposed scheme for position verification by Chakraborty and Leverrier.

Catalytic space: non-determinism and hierarchy

Catalytic computation, defined by Buhrman, Cleve, Koucký, Loff and Speelman (STOC 2014), is a space-bounded computation where in addition to our working memory we have an exponentially larger auxiliary memory which is full; the auxiliary memory may be used throughout the computation, but it must be restored to its initial content by the end of the computation. Motivated by the surprising power of this model, we set out to study the non-deterministic version of catalytic computation. We establish that non-deterministic catalytic log-space is contained in ZPP, which is the same bound known for its deterministic counterpart, and we prove that non-deterministic catalytic space is closed under complement (under a standard derandomization assumption). Furthermore, we establish hierarchy theorems for non-deterministic and deterministic catalytic computation.

Quantum Fully Homomorphic Encryption with Verification

Fully-homomorphic encryption (FHE) enables computation on encrypted data while maintaining secrecy. Recent research has shown that such schemes exist even for quantum computation. Given the numerous applications of classical FHE (zero-knowledge proofs, secure two-party computation, obfuscation, etc.) it is reasonable to hope that quantum FHE (or QFHE) will lead to many new results in the quantum setting. However, a crucial ingredient in almost all applications of FHE is circuit verification. Classically, verification is performed by checking a transcript of the homomorphic computation. Quantumly, this strategy is impossible due to no-cloning. This leads to an important open question: can quantum computations be delegated and verified in a non-interactive manner?

Catalytic Space: Non-determinism and Hierarchy

Catalytic computation, defined by Buhrman, Cleve, Koucký, Loff and Speelman (STOC 2014), is a space-bounded computation where in addition to our working memory we have an exponentially larger auxiliary memory which is full; the auxiliary memory may be used throughout the computation, but it must be restored to its initial content by the end of the computation. Motivated by the surprising power of this model, we set out to study the non-deterministic version of catalytic computation. We establish that non-deterministic catalytic log-space is contained in ZPP, which is the same bound known for its deterministic counterpart, and we prove that non-deterministic catalytic space is closed under complement (under a standard derandomization assumption). Furthermore, we establish hierarchy theorems for non-deterministic and deterministic catalytic computation.