Seyed Sajjad Nezhadi

I am a Computer Science Ph.D. student at QuiCS and the University of Maryland, advised by Matthew Coudron.

Before that I did an undergraduate degree in Math and CS at the University of Toronto, where I was advised by Henry Yuen.

 -  CV  -   - 

Sajjad - سجاد

Research

I am broadly interested in the theory of quantum computing and computational complexity theory.

Quantum Perfect Matching Games

We define the notion of a quantum perfect matching (aswell as fractional matching and X-perfect matching) by defining quantum matching games inspired by the homomorphism game. We prove complexity results aswell as characterization theorems relating these properties to classical properties of graphs aswell as study how these are related to the other quantum graph properties.

The recursive compression method for proving undecidability results

We discuss a technique called recursive compression for proving undecidability results. The technique was used [JNV+22] in the breakthrough MIP* = RE result and further developed [MNY22] to prove the Π2-hardness of the exact quantum value problem. Independently, similar ideas to recursive compression has been used by [DRS10] to prove undecidability results about Wang tilings. We formulate a general recursive compression lemma which abstracts the technique in all these above applications. We also prove a converse and generalizations of the recursive compression lemma.

Hamiltonians whose low-energy states require Ω(n) T gates

In this work we focus on a specific and natural prerequisite to both NLSS and the QPCP Conjecture, namely, the existence of local Hamiltonians whose low-energy states all require ω(log n) T gates to prepare. In fact, we prove a stronger result which is not necessarily implied by either conjecture: we construct local Hamiltonians whose low-energy states require Ω(n) T-gates. We further show that our procedure can be applied to the NLTS Hamiltonians to yield local Hamiltonians whose low-energy states require both Ω(log n)-depth and Ω(n) T gates to prepare. Our results utilize a connection between T-count and stabilizer groups, which was recently applied in the context of learning low T-count states.

Local Hamiltonians with no low-energy stabilizer states

We describe families of local Hamiltonians whose low-energy space contains no stabilizer (Clifford) states via a simple alteration to local Hamiltonians corresponding to CSS codes. Our method can also be applied to the recent NLTS Hamiltonians of Anshu, Breuckmann, and Nirkhe, resulting in a family of local Hamiltonians whose low-energy space contains neither stabilizer states nor trivial states. We hope that our techniques will eventually be helpful for constructing Hamiltonians which simultaneously satisfy NLSS and NLTS.

Nonlocal Games, Compression Theorems, and the Arithmetical Hierarchy

We investigate the connection between the complexity of nonlocal games and the arithmetical hierarchy. We prove that deciding whether the quantum value of a two-player nonlocal game is exactly equal to 1 is complete for Π2. This shows that exactly computing the quantum value is strictly harder than approximating it, and also strictly harder than computing the commuting operator value (either exactly or approximately). We explain how results about the complexity of nonlocal games all follow in a unified manner from a technique known as compression. At the core of our result is a new gapless compression theorem that holds for both quantum and commuting operator strategies. Our compression theorem yields as a byproduct an alternative proof of Slofstra’s result that the set of quantum correlations is not closed.

Synchronous Values of Games

We investigate the synchronous values of graph colouring games, XOR games, and products of games. We show the optimal strategy for a synchronous game need not be synchronous. We derive a formula for the synchronous value of an XOR game as an optimization problem over a spectrahedron and show that the synchronous quantum bias of the XOR of two XOR games is not multiplicative. Finally, we give an example of a game whos repeated product synchronous value is increasing.

On the complexity of zero gap MIP*

We characterize the complexity of exactly computing the maximum winning probability of entangled non-local games and show it to be strictly harder than the halting problem. In particular, we show that the class of zero-gap entangled multiprover interactive proofs, \(MIP_0^*\), is equal to \(\Pi_2\), a class within the second level of arithmetical hierarchy from computability theory.

A generalization of CHSH and the algebraic structure of optimal strategies

We introduce a family of games that generalize CHSH, viewing it as a linear constraint system game. These provide the first example of non-local games that self-test non-Pauli operators and states other than the maximally entangled state. We also introduce the Glued Magic Square game that is not a self-test for any strategy. Our results suggest a richer landscape of self-testing phenomena than previously considered.

Website from Jon Barron via Leonid Keselman's Jekyll fork.