Collective cost of tensor-power state preparation

Problem

Determine the asymptotic weighted circuit cost of preparing tensor powers of a known pure state, and characterize when collective preparation is cheaper per copy than independent preparation. On an arbitrary qubit register, allow Pauli-product rotations \(e^{i\phi P}\) with \(\phi\in\mathbb{R}\) and \(P\in\{I,X,Y,Z\}^{\otimes q}\), assigning each such gate the cost \(|\phi|\). For a known \(m\)-qubit state \(\lvert\psi\rangle\), let \(U(\boldsymbol\phi,\boldsymbol P):=\prod_{j=1}^r e^{i\phi_jP_j}\) for a finite sequence of angles and Pauli products. Define the phase-independent exact \(n\)-copy cost by

\begin{equation} C_n(\psi) :=\inf\left\{ \sum_{j=1}^{r}|\phi_j|: \begin{array}{l} r\in\mathbb N_0,\ \phi_j\in\mathbb R,\ P_j\in\{I,X,Y,Z\}^{\otimes mn}\ (1\le j\le r),\\ U(\boldsymbol\phi,\boldsymbol P) \lvert0^{mn}\rangle\!\langle0^{mn}\rvert U(\boldsymbol\phi,\boldsymbol P)^\dagger =(\lvert\psi\rangle\!\langle\psi\rvert)^{\otimes n} \end{array} \right\}. \tag{1} \end{equation}

Determine the growth of Eq. (1) with \(n\) and characterize the states for which the regularized cost

\begin{equation} C_\infty(\psi) :=\lim_{n\to\infty}\frac{C_n(\psi)}{n} =\inf_{n\ge1}\frac{C_n(\psi)}{n} \tag{2} \end{equation}

satisfies \(C_\infty(\psi)<C_1(\psi)\). For an approximation tolerance \(\varepsilon\ge0\), also determine the scaling of

\begin{equation} C_n^\varepsilon(\psi) :=\inf_U\left\{ \operatorname{cost}(U): \frac12\left\| U\lvert0^{mn}\rangle\!\langle0^{mn}\rvert U^\dagger -(\lvert\psi\rangle\!\langle\psi\rvert)^{\otimes n} \right\|_1\le\varepsilon \right\}, \tag{3} \end{equation}

where \(U\) ranges over all finite circuits of Pauli-product rotations on the \(mn\)-qubit register and \(\operatorname{cost}(U)\) is the corresponding sum of absolute rotation angles, as in Eq. (1). The question for Eq. (3) includes fixed and vanishing error sequences.

Source

The exact and approximate tensor-power preparation questions are posed in the open-problem collection of Krüger and Werner; Scarani et al. independently identify collective product-state preparation complexity as an open direction [KW05], [SIG+05].

Progress

Independent preparation gives \(C_n(\psi)\le nC_1(\psi)\). Concatenation gives \(C_{n+k}(\psi)\le C_n(\psi)+C_k(\psi)\), so Fekete’s lemma establishes the equality in Eq. (2). These elementary bounds do not decide whether the inequality can be strict.
Plesch and Brukner gave nearly optimal worst-case state-synthesis circuits on a fixed register. Their discrete local-gate count neither uses the weighted Pauli-string metric in Eq. (1) nor exploits a tensor-power promise [PB11].
Mora and Briegel related precision-dependent quantum-state algorithmic complexity to entanglement for several state families. Their framework does not determine the direct-product scaling in Eq. (3) [MB05].

Comment

The exact cost model and its approximate variant are posed in the source collection [KW05]; a cloning review also identifies product-preparation complexity as an open direction [SIG+05]. The state is known, so the problem is not universal cloning. The unresolved quantity is the collective saving in Eqs. (2) and (3) for this specific continuous gate metric.

References

[PB11]: M. Plesch and Č. Brukner, “Quantum-State Preparation with Universal Gate Decompositions,” Physical Review A 83, 032302 (2011).DOI arXiv
[MB05]: C. Mora and H. J. Briegel, “Algorithmic Complexity and Entanglement of Quantum States,” Physical Review Letters 95, 200503 (2005).DOI arXiv
[SIG+05]: V. Scarani, S. Iblisdir, N. Gisin, and A. Acín, “Quantum Cloning,” Reviews of Modern Physics 77, 1225–1256 (2005).DOI arXiv
[KW05]: O. Krüger and R. F. Werner (eds.), “Some Open Problems in Quantum Information Theory,” arXiv:quant-ph/0504166 (2005).DOI arXiv

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