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Measurement and computation are fundamental tools for scientific investigation. This

remains true in the quantum realm, even though many of our ordinary intuitions break down — measurements are limited by uncertainty principles, and quantum computers may perform certain calculations exponentially faster than any classical machine. Such quantum effects can nonetheless prove useful, even for these fundamental tasks. Writing in Nature Physics, Seth Lloyd and colleagues1 have proposed a measurement technique that speeds up the process of characterizing an unknown quantum state. The technique, called quantum principal component analysis, or quantum PCA, is exponentially faster than classical methods, but it returns its results in the form of quantum states, rather than classical data. This raises an important question: what does one learn from this kind of procedure?

Broadly speaking, an important goal of experimental quantum information science is to create quantum states of many particles under controlled conditions. Using devices such as ion traps, optical lattices and superconducting qubits, it is becoming possible to manipulate complex quantum states and measure their properties in detail. This allows the investigation of many exotic phenomena, such as high-temperature superconductivity and quantum entanglement between macroscopic objects. It also hints at the tantalizing possibility of building large-scale quantum computers that could solve otherwise intractable problems, such as factoring large integers.

But measuring a large quantum system is a tricky business. A quantum state is described by a density matrix, and the task of estimating this matrix is called quantum state tomography. Quantum state tomography is very resource intensive because the dimensions of the density matrix grow exponentially with the number of particles. In present-day experiments with ten or twenty qubits, one already encounters practical difficulties due to this ‘curse of dimensionality’.

Fortunately, one can use some tricks to mitigate this exponentially growing complexity. Frequently, the quantum states of interest have special features. For example, for thermal states at low temperature and ground states of certain spin systems, the density matrix has a special form, characterized by small matrix- or tensor-rank. For these classes of states, quantum state tomography can be performed more efficiently using techniques such as compressed sensing2–4.

In proposing quantum PCA, Lloyd et al.1

took a radically different approach to the problem of tomography. They asked what could be done if one were allowed to output a quantum state, rather than classical data. Obviously, one should do something that reveals some non-trivial properties of the density matrix — but what?

Lloyd et al.1 considered the case in which the density matrix ρ has rapidly decaying

eigenvalues — as in a low-temperature thermal state. They designed a procedure that estimates the largest eigenvalues of ρ, and outputs the corresponding eigenvectors in the form of quantum states — that is, the output consists of quantum systems, prepared in states corresponding to the eigenvectors of ρ (Fig. 1). This is quantum PCA. The name is a reference to principal component analysis in statistics, where one finds the directions of maximum variance in a high-dimensional data set.

Quantum PCA is very fast: its running time grows polynomially with the number of particles (n), in contrast with the exponential complexity of conventional tomography. This speed-up is possible precisely because each eigenvector is represented using a quantum state of n qubits, rather than a classical vector of dimension 2n. Obtaining the eigenvectors

QUANTUM INFORMATION

Show, don’t tellProbing an unknown quantum state is a resource-intensive endeavour. Now, it is shown that it may be faster to record observations that are themselves quantum superpositions, rather than classical data.

Yi-Kai Liu

Figure 1 | An unknown quantum state can be visualized as an ellipsoid, whose principal axes are the eigenvectors of the density matrix. Quantum principal component analysis (PCA) outputs a collection of quantum states corresponding to the individual axes of the ellipsoid. Each quantum state is drawn inside a circle, to indicate that the information is stored in a quantum superposition. To obtain classical data, one would need to perform measurements on these quantum states.

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in quantum form is both an advantage and a limitation, as it allows the experimenter to measure any observable property of the eigenvectors, but it does not provide a complete classical description of them (unless the experiment is repeated an exponential number of times).

Quantum PCA is built around a simpler operation called density matrix exponentiation. This is a kind of quantum simulation where the dynamics are specified in quantum form. More precisely, given many copies of an unknown quantum state with density matrix ρ, this operation causes another quantum system to evolve in time as if it had a Hamiltonian exactly given by ρ. By combining this trick with the quantum algorithm for phase estimation, one can extract the eigenvectors of ρ, again in quantum form — thus realizing PCA.

Unfortunately, quantum PCA is very challenging to implement experimentally, as it requires extremely accurate two-qubit gates, and its output is a quantum state, which cannot be cloned and is difficult to store for long periods of time. For these reasons, it is probably beyond the capabilities of the current generation of quantum devices.

From a broader perspective, the purpose of quantum PCA is not to replace conventional tomography, but to provide a tool that can be applied to very large quantum systems that are beyond its reach. It is still too early to say exactly how this tool can be put to use. But there are a few hints.

Quantum PCA is potentially useful because it extracts the eigenvectors of the density matrix ρ (a highly nonlinear operation), which reveals information that would not have been immediately accessible using linear measurements of ρ. It is also simpler than other strategies for performing entangled measurements on many copies of ρ, such as the quantum Schur transform5. After running quantum PCA, one can perform a small number of measurements to estimate those quantities that are of physical interest, such as correlation functions or expectation values of local observables.

Looking further into the future, Lloyd et al.1 argue that quantum PCA could be used for large-scale data mining and machine learning. An exponential amount of classical data could be encoded into a quantum state, and then analysed using quantum PCA, provided that the classical data had been stored in an appropriate

quantum device, such as a quantum random-access memory6. Quantum algorithms for solving exponentially large linear systems of equations could also be applied in this scenario7. While this grand vision is unlikely to be realized anytime soon, it illustrates the possible consequences of today’s efforts to build ever more powerful devices for processing quantum information. ❐

Yi-Kai Liu is in the Applied and Computational Mathematics Division at the National Institute of Standards and Technology, Gaithersburg, Maryland 20899, USA. e-mail: yi-kai.liu@nist.gov

References1. Lloyd, S., Mohseni, M. & Rebentrost, P. Nature Phys.

http://dx.doi.org/10.1038/nphys3029 (2014).2. Gross, D., Liu, Y.-K., Flammia, S. T., Becker, S. & Eisert, J.

Phys. Rev. Lett. 105, 150401 (2010).3. Cramer, M. et al. Nature Commun. 1, 149 (2010).4. Shabani, A. et al. Phys. Rev. Lett. 106, 100401 (2011).5. Harrow, A. W. Applications of Coherent Classical Communication

and the Schur Transform to Quantum Information Theory PhD thesis, Massachusetts Institute of Technology (2005).

6. Giovannetti, V., Lloyd, S. & Maccone, L. Phys. Rev. Lett. 100, 160501 (2008).

7. Harrow, A. W., Hassidim, A. & Lloyd, S. Phys. Rev. Lett. 103, 150502 (2009).

Published online: 27 July 2014

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