# Selected Publications

### Quantum Algorithm for Multivariate Polynomial Interpolation

How many quantum queries are required to determine the coefficients of a degree-$d$ polynomial in $n$ variables? We present and analyze quantum algorithms for this multivariate polynomial interpolation problem over the fields $\mathbb{F}_q$, $\mathbb{R}$, and $\mathbb{C}$. We show that $k_{\mathbb{C}}$ and $2k_{\mathbb{C}}$ queries suffice to achieve probability $1$ for $\mathbb{C}$ and $\mathbb{R}$, respectively, where $k_{\mathbb{C}}=\smash{\lceil\frac{1}{n+1}{n+d\choose d}\rceil}$ except for $d=2$ and four other special cases. For $\mathbb{F}_q$, we show that $\smash{\lceil\frac{d}{n+d}{n+d\choose d}\rceil}$ queries suffice to achieve probability approaching $1$ for large field order $q$. The classical query complexity of this problem is $\smash{n+d\choose d}$, so our result provides a speedup by a factor of $n+1$, $\frac{n+1}{2}$, and $\frac{n+d}{d}$ for $\mathbb{C}$, $\mathbb{R}$, and $\mathbb{F}_q$, respectively. Thus we find a much larger gap between classical and quantum algorithms than the univariate case, where the speedup is by a factor of $2$. For the case of $\mathbb{F}_q$, we conjecture that $2k_{\mathbb{C}}$ queries also suffice to achieve probability approaching $1$ for large field order $q$, although we leave this as an open problem.

### The Minimum Size of Unextendible Product Bases in the Bipartite Case (and Some Multipartite Cases)

A long-standing open question asks for the minimum number of vectors needed to form an unextendible product basis in a given bipartite or multipartite Hilbert space. A partial solution was found by Alon and Lovasz in 2001, but since then only a few other cases have been solved. We solve all remaining bipartite cases, as well as a large family of multipartite cases.
Comm. Math. Phys.

### Entanglement Can Completely Defeat Quantum Noise

We describe two quantum channels that individually cannot send any classical information without some chance of decoding error. But together a single use of each channel can send quantum information perfectly reliably. This proves that the zero-error classical capacity exhibits superactivation, the extreme form of the superadditivity phenomenon in which entangled inputs allow communication over zero-capacity channels. But our result is stronger still, as it even allows zero-error quantum communication when the two channels are combined. Thus our result shows a new remarkable way in which entanglement across two systems can be used to resist noise, in this case perfectly. We also show a new form of superactivation by entanglement shared between sender and receiver.
Phys. Rev. Lett.

### Superactivation of the Asymptotic Zero-Error Classical Capacity of a Quantum Channel

The zero-error classical capacity of a quantum channel is the asymptotic rate at which it can be used to send classical bits perfectly so that they can be decoded with zero probability of error. We show that there exist pairs of quantum channels, neither of which individually have any zero-error capacity whatsoever (even if arbitrarily many uses of the channels are available), but such that access to even a single copy of both channels allows classical information to be sent perfectly reliably. In other words, we prove that the zero-error classical capacity can be superactivated. This result is the first example of superactivation of a classical capacity of a quantum channel.
IEEE Trans. Inf. Theory

# Recent Publications

• A Separability-Entanglement Classifier via Machine Learning

• Quantum Algorithm for Multivariate Polynomial Interpolation

• A Finite Presentation of CNOT-dihedral Operators

• Local Density Matrices of Many-body States in the Constant Weight Subspaces

• Joint Product Numerical Range and Geometry of Reduced Density Matrices

• Quantum State Tomography via Reduced Density Matrices

• Quantifying the Coherence of Pure Quantum States

• Pure State Tomography with Pauli Measurements

• Tomography is Necessary for Universal Entanglement Detection with Single-copy Observables

• Detecting Consistency of Overlapping Quantum Marginals by Separability

# Projects

#### External Project

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#### Machine Learning for Quantum Physics

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# Teaching

I was a teaching instructor for the following course at University of Guelph

• MATH2150DE Applied Matrix Algebra (Winter 2014)

I was a teaching instructor for the following course at University of Waterloo

• QIC890 and 891 Selected Advanced Topics in Quantum Information (Spring 2012, Course Coordinator: Michele Mosca)

I was a teaching assistant for the following courses at Tsinghua University:

• Discrete Mathematics (2007)
• Discrete Mathematics (2006)

# Contact

• jchen@ios.ac.cn
• 4 Zhongguancun South 4th St, Haidian Qu, Beijing Shi, China, 100080