Quantum Data Compression of a Qubit Ensemble

Data compression is a ubiquitous aspect of modern information technology, and the advent of quantum information raises the question of what types of compression are feasible for quantum data, where it is especially relevant given the extreme difficulty involved in creating reliable quantum memories. We present a protocol in which an ensemble of quantum bits (qubits) can in principle be perfectly compressed into exponentially fewer qubits. We then experimentally implement our algorithm, compressing three photonic qubits into two. This protocol sheds light on the subtle differences between quantum and classical information. Furthermore, since data compression stores all of the available information about the quantum state in fewer physical qubits, it could allow for a vast reduction in the amount of quantum memory required to store a quantum ensemble, making even today’s limited quantum memories far more powerful than previously recognized.

Figure 1

(a) Quantum Schur-Weyl transform. A three-qubit quantum Schur-Weyl transform. ${\widehat{U}}_1$ and ${\widehat{U}}_2$ are unitaries (whose detailed descriptions can be found in the main text) which are controlled by the upper qubit; the H is a controlled Hadamard and the other two-qubit gates are cnots, while the three-qubit gate is a Toffoli. (b) Simplified circuit. The shaded area labeled 2 can be viewed as a two-qubit unitary gate, $\widehat{A}$, acting on the first two qubits which is controlled by the third qubit. C is a two-qubit unitary gate which is applied (or not) based on a measurement of qubit 3. The numbered boxes correspond to the areas in Fig. 2 which show the physical implementation of the circuit elements.

Figure 2

Optical Implementation. (a)–(b) State preparation and data compression. Two photons, generated via spontaneous parametric down-conversion (SPDC), are used to encode three qubits. Qubit 1 is encoded in the polarization of photon 1, qubit 2 in its path degree-of-freedom (the logical paths are labeled $P_0$ and $P_1$), and qubit 3 in the polarization of photon 2 (initially entangled with an additional path degree of freedom of photon 1). After data compression, only photon 1 remains, encoding a path and polarization qubit. (c)–(d) Measuring the compressed qubits. Any single-qubit measurement can be made on the compressed state in two steps by first setting the basis (c), and then measuring $\widehat{Z}$ (d). The $\widehat{Z}$ measurement has four outcomes: $H{P}_1$, $V{P}_1$, $H{P}_0$, and $V{P}_1$ [where $H$ ($V$) stands for horizontal (vertical) polarization]. The areas numbered 1–3 correspond to circuit elements shown in Fig. 1.

Figure 3

Sample raw data for an input state $\cos (2\theta )|0⟩+\sin (2\theta )|1⟩$, for $\theta =13.5°$. (a)–(c) Histograms of estimates the spin along $\widehat{Z}$, $\widehat{Y}$, and $\widehat{X}$, after $M$ trials (defined in the text) of the data compression circuit. The bars are experimentally measured data, and the blue (red) curve is a normal distribution of width $V_1/3M$ ($V_1/2M$) normalized to have the same area as the experimental histogram, where $V_1$ is the single-qubit variance. (d)–(f) The experimentally observed probabilities for measuring the two qubits and finding them in $|00⟩$, $|01⟩$, $|10⟩$, or $|11⟩$ for $\widehat{Z}$, $\widehat{Y}$, and $\widehat{X}$ measurements. The dark blue bars are the experimentally measured counts, normalized by the total number of counts, and the light bars are the theoretically predicted results.

Figure 4

Measurement variances for various input states. The solid blue lines are the theoretical variances resulting from performing a measurement on three independent qubits (and thus our two compressed qubits), the dashed red lines are for two independent qubits, the grey dashed lines are the theoretical variance when two independent qubits are measured optimally and a random measurement is performed on a third qubit, and the circles are the variances which we observe when experimentally measuring the two compressed qubits. (a)–(c) By sending in various input states, parameterized as $\cos 2\theta |0⟩+\sin 2\theta |1⟩$, we see that the two compressed qubits demonstrate the statistics of three independent qubits for $\widehat{Z}$, $\widehat{Y}$, and $\widehat{X}$ measurements. (d) Averaging the above variances yields the variance averaged over all possible measurements.