Second Quantized Kolmogorov Complexity

Scarani

2015

Annalen der Physik

The complexity of a quantum state may be closely related to the usefulness of the state for quantum computation. We discuss this link using the tree size of a multiqubit state, a complexity measure that has two noticeable (and, so far, unique) features: it is in principle computable, and non-trivial lower bounds can be obtained, hence identifying truly complex states. In this paper, we first review the definition of tree size, together with known results on the most complex three and four qubits states. Moving to the multiqubit case, we revisit a mathematical theorem for proving a lower bound on tree size that scales superpolynomially in the number of qubits. Next, states with superpolynomial tree size, the Immanant states, the Deutsch-Jozsa states, the Shor's states and the subgroup states, are described. We show that the universal resource state for measurement based quantum computation, the 2D-cluster state, has superpolynomial tree size. Moreover, we show how the complexity of subgroup states and the 2D cluster state can be verified efficiently. The question of how tree size is related to the speed up achieved in quantum computation is also addressed. We show that superpolynomial tree size of the resource state is essential for measurement based quantum computation. The necessary role of large tree size in the circuit model of quantum computation is still a conjecture; and we prove a weaker version of the conjecture.

Section: Introductionmentioning

confidence: 99%

State complexity and quantum computation

Scarani

2015

Annalen der Physik

“…Among the different measures of complexity, quantum Kolmogorov complexity is the attempt to quantify complexity of states in the most general way [4][5][6]; however, this measure suffers from the setback that it is not computable and only upper bounds can be given. Therefore it is possible to certify that a state is not complex, but it is impossible to certify that a state is complex.…”

Section: Introductionmentioning

confidence: 99%

Maximal tree size of few-qubit states

et al. 2014

Phys. Rev. A

Tree size (TS) is an interesting measure of complexity for multiqubit states: not only is it in principle computable, but one can obtain lower bounds for it. In this way, it has been possible to identify families of states whose complexity scales superpolynomially in the number of qubits. With the goal of progressing in the systematic study of the mathematical property of TS, in this work we characterize the tree size of pure states for the case where the number of qubits is small, namely, 3 or 4. The study of three qubits does not hold great surprises, insofar as the structure of entanglement is rather simple; the maximal TS is found to be 8, reached for instance by the |W state. The study of four qubits yields several insights: in particular, the most economic description of a state is found not to be recursive. The maximal TS is found to be 16, reached for instance by a state called |Ψ (4) which was already discussed in the context of four-photon down-conversion experiments. We also find that the states with maximal tree size form a set of zero measure: a smoothed version of tree size over a neighborhood of a state ( − TS) reduces the maximal values to 6 and 14, respectively. Finally, we introduce a notion of tree size for mixed states and discuss it for a one-parameter family of states.

“…We believe that an explicit construction will be very useful for further studies on the topic of complex quantum states. Among the several complexity measures proposed [16,[19][20][21][22][23], we focus on the tree size of a quantum state, introduced by Aaronson in an attempt to give a more rigorous foundation to the debate on the possibility of large-scale quantum computing versus a hypothetical breakdown of quantum mechanics [16]. This measure of complexity is motivated by the work of Raz, who showed that any multilinear formula for the determinant and permanent of a matrix must be superpolynomial in size [24].…”

Section: Introductionmentioning

confidence: 99%

Tree-size complexity of multiqubit states

Nguyen

et al. 2013

Phys. Rev. A

Complexity is often invoked alongside size and mass as a characteristic of macroscopic quantum objects. In 2004, Aaronson introduced the \textit{tree size} (TS) as a computable measure of complexity and studied its basic properties. In this paper, we improve and expand on those initial results. In particular, we give explicit characterizations of a family of states with superpolynomial complexity $n^{\Omega(\log n)}= \mathrm{TS} =O(\sqrt{n}!)$ in the number of qubits $n$; and we show that any matrix-product state whose tensors are of dimension $D\times D$ has polynomial complexity $\mathrm{TS}=O(n^{\log_2 2D})$.Comment: 7 pages, 2 figure