【Abstract】Quantum state tomography is a technique for determining an unknown quantum state. One approach to implementing it is through a Positive Operator-Valued Measure (POVM). In this method, many identical copies of an unknown quantum state are prepared and measured to reconstruct the state. A POVM provides a more general framework for quantum measurement than the standard projective measurement on an orthonormal basis. In this article, we implement a SIC-POVM, one particular class of POVMs, for a single qubit on an IBM Quantum processor (ibm_kingston). The experimental results demonstrate that several unknown quantum states can be reconstructed with high accuracy.
🟢 Overview of SIC-POVM
SIC-POVM stands for "Symmetric Informationally Complete Positive Operator-Valued Measure". The general theory of POVMs is described in References [1], [2], and [3], while Reference [3] provides a detailed discussion of SIC-POVMs. For a quantum system of dimension d, a SIC-POVM consists of d2 projection operators,
Ei = |ψi><ψi|/d
and satisfies the following formula:
|<ψi|ψj>|2 = 1/(d+1) (i ≠ j)
|<ψi|ψj>|2 = 1/(d+1) (i ≠ j)
Here, |ψi> represents a pure quantum state on the Bloch sphere.
For a single qubit (d = 2), the SIC-POVM consists of four state vectors. The squared inner product between any pair of distinct vectors is therefore 1/3. A specific example is shown in Fig.1. For convenience, it is customary to choose the first state as |ψ1> = |0>, and the four vertices corresponding to these state vectors form a geometrically elegant regular tetrahedron inscribed in the Bloch sphere.
🟢 Implementing Quantum State Tomography
An unknown quantum state cannot be reconstructed using only the conventional Z-basis measurement. For example, if measurements along the Z axis yield |0> and |1> with equal probabilities of 50%, multiple quantum states are consistent with those results. Additional measurements along the X and Y axes are therefore required.
A SIC-POVM, in contrast, can be regarded as a special measurement device with four output channels. When a large number of identical copies of an unknown qubit are sent into this device, each output channel records a certain number of detection events. The relative frequencies (probability distribution) represent the overlap between the unknown quantum state and each of the four SIC-POVM state vectors. If the state corresponding to the i-th SIC-POVM element is denoted by |ψi>, the measurement probability becomes as follows:
Pi = (1/2)|<ψi|Unknown state >|2
A particularly attractive feature of SIC-POVM tomography is that these four probabilities directly determine the Cartesian coordinates (X,Y,Z) of the unknown state on the Bloch sphere through the following equations:
X = √2(2P2 - P3 - P4)
Y = √6(P3 - P4)
Z = 4P1 - 1
🟢 Performing SIC-POVM Measurements on an IBM Quantum
If a known quantum state is simply treated as an unknown one, reconstructing it using the SIC-POVM formulas presents no theoretical difficulty. Implementing the procedure on an actual quantum computer or a quantum circuit simulator, however, requires some additional consideration.
On a real quantum processor, the probability is not obtained by directly calculating the inner product between the state vector and a measurement vector. Instead, many identical copies of the quantum state are prepared and measured repeatedly. The measurement probabilities are then estimated from the observed frequencies according to Born's rule.
Most quantum computers, however, can perform only projective measurements in the computational (Z) basis. As shown in Fig. 1, three of the four SIC-POVM measurement directions point diagonally toward the southern hemisphere of the Bloch sphere rather than along the Z axis.
To overcome this limitation, an appropriate unitary rotation is applied immediately before measurement so that the desired SIC-POVM measurement direction is mapped onto the Z axis. A standard Z-basis measurement is then performed, yielding the measurement probability corresponding to that SIC-POVM element. The present experiment follows exactly this procedure.
🟢 Two Quantum States Treated as Unknown
In our experiment, the two known quantum states shown in Fig. 2 were deliberately treated as if they were unknown. Both are pure states located on the surface of the Bloch sphere. In Case 1, the state lies on the equator with a phase angle of 0. In Case 2, the state is located in the southern hemisphere with a phase angle of π/3. We expected that the state in Case 1 would be easier to reconstruct than that in Case 2.
It should be emphasized that the parameters used to define these "unknown" states (θ and φ) were not used anywhere in the reconstruction process. Only the four SIC-POVM measurement directions were used. Consequently, the same procedure would work equally well for a genuinely unknown input quantum state.
🟢 Reconstruction Results
Table1 summarizes the results obtained for both cases using the Qiskit simulator and the IBM Quantum processor (ibm_kingston). The table lists the Cartesian coordinates (X,Y,Z) of the original quantum states treated as unknown, together with the measured probabilities (P1, P2, P3, P4) obtained from both the simulator and the real quantum hardware.
The reconstructed Cartesian coordinates (X, Y, Z) deserve particular attention. As expected, the quantum state in Case 1 was reproduced almost perfectly by the IBM Quantum processor. The state in Case 2 was also reconstructed with a high degree of accuracy, demonstrating that SIC-POVM-based quantum state tomography can successfully identify unknown single-qubit states even on present-day quantum hardware.
【References on POVM】
[1] Peter Y. Lee, Huiwen Ji, Ran Cheng, "Quantum Computing and Information", 2nd Edition, Polaris QCI Publishing, 2025.
(especially, 3.5 Application to Quantum State Tomography, and Chapter12(Density Operators and Quantum Channels))
[2] Michael A. Nielsen and Isaac L. Chuang, "Quantum Computation and Quantum Information", Version 13, Cambridge University Press, 2023.
(2.2.6 POVM measurements)
[3] John Watrous, "General formulation of quantum information" in IBM Quantum Platform Learning.
https://quantum.cloud.ibm.com/learning/en/courses/general-formulation-of-quantum-information/general-measurements/introduction
https://quantum.cloud.ibm.com/learning/en/courses/general-formulation-of-quantum-information/general-measurements/introduction
