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Abstract

After Bob sends Alice a bit, she responds with a lengthy reply. At the cost of a factor of two in the total communication, Alice could just as well have given the two possible replies without listening and have Bob select which applies to him. Motivated by a conjecture stating that this form of "round elimination" is impossible in exact quantum communication complexity, we study the orthogonal rank and a symmetric variant thereof for a certain family of Cayley graphs. The orthogonal rank of a graph is the smallest number d for which one can label each vertex with a nonzero d-dimensional complex vector such that adjacent vertices receive orthogonal vectors. We show an exp(n) lower bound on the orthogonal rank of the graph on {0, 1} n in which two strings are adjacent if they have Hamming distance at least n/2. In combination with previous work, this implies an affirmative answer to the above conjecture.

Quantum singular value transformation and beyond: exponential improvements for quantum matrix arithmetics

“…In the special case when one has access to the square root of H, and H has an eigenvalue close to 0, then one can still achieve quadratically improved scaling with β as shown by Chowdhury and Somma [CS17]. This can be easily shown using our techniques observing that e −βH = e −β( √ H) 2 , and that the function e −βx 2 can be ε-approximated on the interval [0, 1] using a polynomial of degree O √ β log 1 ε as follows from Theorem 63 or Corollary 64.…”

Quantum SDP-Solvers: Better upper and lower bounds

“…Also, our purified Gibbs sampler has logarithmic dependence on the error, which is exponentially better than the Gibbs sampler of Poulin and Wocjan [PW09b] that Brandão and Svore invoke. Chowdhury and Somma [CS17] also gave a Gibbs sampler with logarithmic error-dependence, but assuming query access to the entries of √ H rather than H itself.…”

“…Our Gibbs sampler uses similar methods to the work of Chowdhury and Somma [CS17] for achieving logarithmic dependence on the precision. However, the result of [CS17] cannot be applied to our setting, because it implicitly assumes access to an oracle for √ H instead of H. Although their paper describes a way to construct such an oracle, it comes with a large overhead: they construct an oracle for √ H = √ H + νI, where ν ∈ R + is some possibly large positive number. This shifting can have a huge effect on Z = Tr e −H = e −ν Tr e −H , which can be prohibitive due to the 1/Z factor in the runtime, which blows up exponentially in ν.…”

“…This approach was already used in several earlier papers, particularly in [CKS17,CS17]. There the main technical difficulty was to obtain a good truncated Fourier series.…”

Optimising Matrix Product State Simulations of Shor's Algorithm

“…Instead, by using the matrix product state representation [15] of a quantum state, the space requirements scale according to the amount of entanglement present in the system. As tensor networks [16], matrix product states were originally used for simulating one-dimensional quantum many-body systems [17,18], but have since been adapted for simulating quantum circuits [14,15,19]. As such, even states of many qubits may feasibly be stored using a matrix product state representation, provided its entanglement is sufficiently limited.…”