Deterministic quantum computation with one quantum bit (DQC1), or the one clean qubit model, [E. Knill and R. Laflamme, Phys. Rev. Lett. {\bf81}, 5672 (1998)] is a model of quantum computing where the input is the tensor product of a single pure qubit and many completely-mixed states, and only the single qubit is measured at the end of the computation. In spite of its naive appearance, the DQC1 model can efficiently solve some problems for which no classical efficient algorithms are known, and therefore it has been conjectured that the DQC1 model is more powerful than classical computing (under the assumption of $\mbox{BPP}\subsetneq\mbox{BQP}$). Here we show that the output probability distribution of the DQC1 model cannot be classically efficiently approximated (exactly within a polynomial bit length or in the fully polynomial randomized approximation scheme (FPRAS) with at most a constant error) unless $\mbox{BQP}\subseteq\mbox{BPP}$.
Deterministic quantum computation with one quantum bit (DQC1), or the one clean qubit model, [E. Knill and R. Laflamme, Phys. Rev. Lett. {\bf81}, 5672 (1998)] is a model of quantum computing where the input is the tensor product of a single pure qubit and many completely-mixed states, and only the single qubit is measured at the end of the computation. In spite of its naive appearance, the DQC1 model can efficiently solve some problems for which no classical efficient algorithms are known, and therefore it has been conjectured that the DQC1 model is more powerful than classical computing (under the assumption of \mboxBPP⊊\mboxBQP). Here we show that the output probability distribution of the DQC1 model cannot be classically efficiently approximated (exactly within a polynomial bit length or in the fully polynomial randomized approximation scheme (FPRAS) with at most a constant error) unless \mboxBQP⊆\mboxBPP.
Deterministic quantum computation with one quantum bit (DQC1) [E. Knill and R. Laflamme, Phys. Rev. Lett. 81, 5672 (1998)] is a model of quantum computing where the input is restricted to containing a single qubit in a pure state and has all other qubits in a completely mixed state. Only the single pure qubit is measured at the end of the computation. While it is known that DQC1 can efficiently solve several problems for which no known classical efficient algorithms exist, the question of whether DQC1 is really more powerful than classical computation remains open. In this Letter, we introduce a slightly modified version of DQC1, which we call DQC1(k), where k output qubits are measured, and show that DQC1(k) cannot be classically efficiently simulated for any k≥3 unless the polynomial hierarchy collapses at the third level.
The one-clean-qubit model (or the deterministic quantum computation with one quantum bit model) is a restricted model of quantum computing where all but a single input qubits are maximally mixed. It is known that the probability distribution of measurement results on three output qubits of the one-clean-qubit model cannot be classically efficiently sampled within a constant multiplicative error unless the polynomial-time hierarchy collapses to the third level [T. Morimae, K. Fujii, and J. F. Fitzsimons, Phys. Rev. Lett. 112, 130502 (2014)PRLTAO0031-900710.1103/PhysRevLett.112.130502]. It was open whether we can keep the no-go result while reducing the number of output qubits from three to one. Here, we solve the open problem affirmatively. We also show that the third-level collapse of the polynomial-time hierarchy can be strengthened to the second-level one. The strengthening of the collapse level from the third to the second also holds for other subuniversal models such as the instantaneous quantum polynomial model [M. Bremner, R. Jozsa, and D. J. Shepherd, Proc. R. Soc. A 467, 459 (2011)PRLAAZ1364-502110.1098/rspa.2010.0301] and the boson sampling model [S. Aaronson and A. Arkhipov, STOC 2011, p. 333]. We additionally study the classical simulatability of the one-clean-qubit model with further restrictions on the circuit depth or the gate types.
The one-clean qubit model (or the DQC1 model) is a restricted model of quantum computing where only a single qubit of the initial state is pure and others are maximally mixed. Although the model is not universal, it can efficiently solve several problems whose classical efficient solutions are not known. Furthermore, it was recently shown that if the one-clean qubit model is classically efficiently simulated, the polynomial hierarchy collapses to the second level. A disadvantage of the one-clean qubit model is, however, that the clean qubit is too clean: for example, in realistic NMR experiments, polarizations are not high enough to have the perfectly pure qubit. In this paper, we consider a more realistic one-clean qubit model, where the clean qubit is not clean, but depolarized. We first show that, for any polarization, a multiplicative-error calculation of the output probability distribution of the model is possible in a classical polynomial time if we take an appropriately large multiplicative error. The result is in strong contrast with that of the ideal one-clean qubit model where the classical efficient multiplicative-error calculation (or even the sampling) with the same amount of error causes the collapse of the polynomial hierarchy. We next show that, for any polarization lower-bounded by an inverse polynomial, a classical efficient sampling (in terms of a sufficiently small multiplicative error or an exponentially small additive error) of the output probability distribution of the model is impossible unless BQP (bounded error quantum polynomial time) is contained in the second level of the polynomial hierarchy, which suggests the hardness of the classical efficient simulation of the one nonclean qubit model.
The one-clean-qubit model (or the DQC1 model) is a restricted model of quantum computing where only a single input qubit is pure and all other input qubits are maximally mixed. In spite of the severe restriction, the model can solve several problems (such as calculating Jones polynomials) whose classical efficient solutions are not known. Furthermore, it was shown that if the output probability distribution of the one-clean-qubit model can be classically efficiently sampled with a constant multiplicative error, then the polynomial hierarchy collapses to the second level. Is it possible to improve the multiplicative error hardness result to a constant total variation distance error one like other subuniversal quantum computing models such as the IQP (Instantaneous Quantum Polynomial time) model, the boson sampling model, and the Fourier sampling model? In this paper we show that it is indeed possible if we accept a modified version of the average case hardness conjecture. Interestingly, the anticoncentration lemma can be easily shown by using the special property of the one-clean-qubit model that each output probability is so small that no concentration occurs.
I hereby declare that I am the sole author of this thesis. This is a true copy of the thesis, including any required final revisions, as accepted by my examiners. I understand that my thesis may be made electronically available to the public. G. Passante ii Quantum information processors have the ability to drastically change our world. By manipulating bits of information ruled by the laws of quantum mechanics, computational devices can perform some computations that are classically intractable. Most quantum algorithms rely on pure qubits as inputs and require entanglement throughout the compu-tation. In this thesis, we explore a model of computation that uses mixed qubits without en-tanglement known as DQC1 (deterministic quantum computation with one quantum bit), using the physical system of liquid-state Nuclear Magnetic Resonance (NMR). Throughout our research, we experimentally implement an algorithm that completely encapsulates the DQC1 model, and take a close look at the quantum nature of DQC1-states as given by the
Publicly accessible fault-tolerant quantum computers must preserve scientific utility while limiting cryptanalytic power. We propose the restricted model of computation, 1/2BQP_1: a quantum server provides random computational-basis inputs, revealed only after execution, and one designated output bit. This interface permits arbitrary circuits and system sizes. We conjecture that a classical client making polynomially many adaptive requests cannot efficiently factor RSA moduli. The model subsumes the well-known one-clean-qubit model DQC1, retaining its applications to infinite-temperature multi-time correlations, out-of-time-order correlators, and suitably normalized partition functions. We propose a candidate for separating 1/2BQP_1 from DQC1 based on testing classical predictions of quantum spin dynamics. We argue security in two steps. First, we show that one-bit readout makes the quantum stages of standard factoring algorithms classically simulable, even when their preparation and postprocessing are redesigned within a broad family around a single arithmetic-oracle call. This covers the quantum stages of Shor, Ekera-Hastad, Kitaev, and Regev factoring constructions. Second, we examine the workaround of coherently implementing rational reconstruction to release a factor bit. Random inputs obstruct this straightforward attack: known techniques accommodate logarithmic-depth classical circuits, but rational reconstruction has resisted such parallelization for decades. Therefore, we show that security of our protocol is built on not only classical hardness for factoring, but also conjectured circuit lowerbounds for rational reconstruction.